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+Trade-offs between cost and information in cellular prediction
+Age J. Tjalma,1 Vahe Galstyan,1 Jeroen Goedhart,1 Lotte Slim,1 Nils B. Becker,2 and Pieter Rein ten Wolde1, ∗
+1AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
+2Theoretical Systems Biology, German Cancer Research Center, 69120 Heidelberg, Germany
+(Dated: January 11, 2023)
+Living cells can leverage correlations in environmental fluctuations to predict the future environ-
+ment and mount a response ahead of time. To this end, cells need to encode the past signal into the
+output of the intracellular network from which the future input is predicted. Yet, storing information
+is costly while not all features of the past signal are equally informative on the future input signal.
+Here, we show, for two classes of input signals, that cellular networks can reach the fundamental
+bound on the predictive information as set by the information extracted from the past signal: push-
+pull networks can reach this information bound for Markovian signals, while networks that take
+a temporal derivative can reach the bound for predicting the future derivative of non-Markovian
+signals. However, the bits of past information that are most informative about the future signal are
+also prohibitively costly. As a result, the optimal system that maximizes the predictive information
+for a given resource cost is, in general, not at the information bound. Applying our theory to the
+chemotaxis network of Escherichia coli reveals that its adaptive kernel is optimal for predicting
+future concentration changes over a broad range of background concentrations, and that the system
+has been tailored to predicting these changes in shallow gradients.
+Keywords: prediction, information bottleneck, sensing, resource allocation
+Single-celled organisms live in a highly dynamic envi-
+ronment to which they continually have to respond and
+adapt.
+To this end, they employ a range of response
+strategies, tailored to the temporal structure of the envi-
+ronmental variations. When these variations are highly
+regular, such as the daily light variations, it becomes ben-
+eficial to develop a clock from which the time and hence
+the current and future environment can be inferred [1, 2].
+In the other limit, when the fluctuations are entirely un-
+predictable, cells have no choice but to resort to either
+the strategy of detect-and-respond or the bet-hedging
+strategy of stochastic switching between different phe-
+notypes [3]. Yet arguably the most fascinating strategy
+lies in between these two extremes. When the environ-
+mental fluctuations happen with some regularity, then it
+becomes feasible to predict the future environment and
+initiate a response ahead of time. While it is commonly
+believed that only higher organisms can predict the fu-
+ture, experiments have vividly demonstrated that even
+single-cell organisms can leverage temporal correlations
+in environmental fluctuations in order to predict, e.g.,
+future nutrient levels [4, 5].
+The ability to predict future signals can provide a fit-
+ness benefit [6]. The capacity to anticipate changes in
+oxygen levels [4], or the arrival of sugars or stress signals
+[5], can increase the growth rate of single-celled organ-
+isms; modeling has revealed that prediction can enhance
+bacterial chemotaxis [7].
+Yet, a predict-and-anticipate
+strategy is only advantageous if the cell can reliably pre-
+dict the future on timescales that are longer than the
+time it takes to mount a response. What fundamentally
+limits the accuracy of cellular prediction remains, how-
+ever, poorly understood.
+∗ p.t.wolde@amolf.nl
+While the cell needs to predict the future environ-
+ment, it can only sense the present and remember the
+past (Fig. 1A). Consequently, for a given amount of in-
+formation the cell can store about the present and past
+signal, there is a maximum amount of information it can
+possibly have about the future [6, 8] (Fig. 1C-I). This in-
+formation bound is determined by the temporal structure
+of the environmental fluctuations [8, 9].
+How close cells can come to this bound depends on
+the design of the intracellular biochemical network that
+senses and processes the environmental signals (Fig. 1B).
+To maximize the predictive power the cell must use its
+memory effectively: it should extract only those charac-
+teristics from the present and past signal that are most
+informative about the future [7]. Whether it can do so,
+is determined by the topology of the signaling network.
+Moreover, like any information processing device, bio-
+chemical networks require resources to be built and run.
+Molecular components are needed to construct the net-
+work, space is required to accommodate the components,
+time is needed to process the information, and energy is
+required to synthesize the components and operate the
+network [10]. These resources constrain the design and
+performance of any biochemical network, and the ca-
+pacity to sense and process information is no exception
+(Fig. 1C-II).
+Cellular signaling systems provide a unique opportu-
+nity for revealing the resource requirements for predic-
+tion. Cells live in a highly dynamic environment, with
+temporal statistics that are expected to vary markedly.
+Moreover, signaling networks have distinct topologies,
+which are likely tailored to the temporal statistics of the
+environment [7]. In addition, for cellular systems we can
+actually quantify the information processing capacity as
+a function of the resources that are necessary to build
+and run them—protein copies, time, and energy [10, 11].
+arXiv:2301.03964v1  [physics.bio-ph]  10 Jan 2023
+
+2
+Cellular systems are thus ideal for elucidating the rela-
+tionships between future and past information, system
+design (i.e. network topology) and resource constraints.
+Here, we derive the bound on the prediction precision as
+set by the information extracted from the past signal for
+two types of input signals. We will determine how close
+cellular networks can come to this bound, and how this
+depends on the topology of the network and the resources
+to build and run it.
+We find that for the two classes of input signals stud-
+ied, cellular networks exists that can reach the informa-
+tion bound, yet reaching the bound is exceedingly costly.
+The first class of input signals consists of Markovian sig-
+nals. Using the Information Bottleneck Method (IBM)
+[8, 12], we first show that the system that reaches the
+information bound copies the most recent input signal
+into the output from which the future input is predicted.
+Push-pull networks consisting of chemical modification or
+GTPase cycles, which are ubiquitous in prokaryotic and
+eukaryotic cells [13, 14], should be able to reach the infor-
+mation bound, because they are at heart copying devices
+[10, 11]. Yet, copying the most recent input into the out-
+put is extremely costly, because the operating cost, as set
+by the chemical power to drive the cycle, diverges at high
+copying speed. More surprisingly, our results show that
+the predictive and past information can be raised simul-
+taneously by moving away from the information bound,
+even when the operating cost is negligible: the optimal
+system that maximizes the predictive information for a
+given protein synthesis cost is, in general, not at the in-
+formation bound. The number of bits of past information
+per protein cost can be raised by increasing the integra-
+tion time. While this decreases the predictive power per
+bit of past information, thereby moving the system away
+from the information bound, it can increase the total pre-
+dictive information per protein cost. Our analysis thus
+highlights that not all bits of past information are equally
+costly, nor predictive.
+Living cells that navigate their environment typically
+experience signals with persistence as generated by their
+own motion, which motivated us to study a simple class
+of non-Markovian signals. Moreover, these cells can typ-
+ically detect changes in the concentration over a range of
+background concentrations that is orders of magnitude
+larger than the change in the concentration over the ori-
+entational correlation time of their movement. Our anal-
+ysis reveals that in such a scenario the optimal kernel that
+allows the system to reach the information bound on pre-
+dicting the future input derivative is a perfectively adap-
+tive, derivative-taking kernel, precisely as the bacterium
+E. coli employs [15]. We again find, however, that reach-
+ing the information bound is prohibitively costly. The
+reason is that taking an instantaneous derivative, which
+is the characteristic of the input that is most informative
+about the future derivative, reduces the gain to zero be-
+cause the system instantly adapts; the response becomes
+thwarted by biochemical noise. The optimal system that
+maximizes the predictive information under a resource
+constraint thus emerges from a trade-off between taking a
+derivative that is recent and one that is reliable. Finally,
+our analysis reveals that the E. coli chemotaxis system
+has been optimally designed to predict future concentra-
+tion changes in shallow gradients.
+RESULTS
+We focus on cellular signaling systems that respond
+linearly to changes in the input signal [11, 16–19]. These
+systems not only allow for analytical results, but also
+describe information transmission often remarkably well
+[19–22]. The output of these systems can be written as
+x(t) =
+� t
+−∞
+dt′k(t − t′)ℓ(t′) + ηx(t),
+(1)
+where k(t) is the linear response function, ℓ(t) the input
+signal, and ηx(t) describes the noise in the output. We
+will consider stationary signals with different temporal
+correlations, obeying Gaussian statistics.
+Any prediction about the future state of the environ-
+ment must be based on information obtained from its
+past (Fig. 1C-I). In particular, the cell needs to predict
+the input ℓτ ≡ ℓ(t + τ) at a time τ into the future from
+the current output x0 ≡ x(t), which itself depends on
+the input signal in the past, Lp ≡ (ℓ(t), ℓ(t′), · · · ), with
+t > t′ > · · · . The (qualitative) shape of the integration
+kernel k(t), e.g. exponential, adaptive or oscillatory, is
+determined by the topology of the signaling network [7].
+The kernel shape describes how the past signal is mapped
+onto the current output, and hence which characteristics
+of the past signal the cell uses to predict the future signal.
+To maximize the accuracy of prediction, the cell should
+extract those features that are most informative about
+the future signal. These depend on the statistics of the
+input signal.
+Deriving the upper bound on the predictive informa-
+tion as set by the past information is an optimisation
+problem, which can be solved using the IBM [8]. It en-
+tails the maximization of an objective function L:
+max
+P (x0|Lp) [L ≡ I(x0; ℓτ) − γI(x0; Lp)] .
+(2)
+Here, Ipred ≡ I(x0; ℓτ) is the predictive information,
+which is the mutual information between the system’s
+current output x0 and the future ligand concentration
+ℓτ. The past information Ipast ≡ I(x0; Lp) is the mutual
+information between x0 and the trajectory of past lig-
+and concentrations Lp. The Lagrange multiplier γ sets
+the relative cost of storing past over obtaining predic-
+tive information. Given a value of γ, the objective func-
+tion in Eq. 2 is maximized by optimizing the conditional
+probability distribution of the output given the past in-
+put trajectory, P(x0|Lp). For the linear systems consid-
+ered here, this corresponds to optimizing the mapping
+of the past input signal onto the current output via the
+
+3
+past information
+predictive information
+resources
+inaccessible
+inaccessible
+I
+II
+information bound
+A
+B
+C
+Different input signals
+maintenance 
+operating
+Optimal networks
+X
+X
+RL
+*
+time
+signal
+output
+past info
+predictive info
+now
+concentration
+FIG. 1. Cells use biochemical networks to remember the past and predict the future. (A) Cells compress the past
+input into the dynamics of the signalling network from which the future input is then predicted. (B) The optimal topology
+of the network for predicting the future signal depends on the temporal statistics of the input signal. Push-pull networks,
+consisting of chemical modification cycles or GTPase cycles, can optimally predict the future value of Markovian signals, with
+correlation time τℓ; derivative-taking networks, like the E. coli chemotaxis system, can optimally predict the future derivative
+of non-Markovian signals, with correlation time τv. The push-pull network consists of a receptor that drives a downstream
+phosphorylation cycle.
+The ligand binds the receptor with a correlation time τc.
+The push-pull network, driven by ATP
+turnover, integrates the receptor with an integration time τr. The chemotaxis system is a push-pull network, yet augmented
+with negative feedback on the receptor activity via methylation on a timescale τm, as indicated by the dashed grey line. The
+total resource cost consists of a maintenance cost of receptor and readout synthesis at the growth rate λ, and an operating
+cost of driving the cycle. (C) The predictive information on the future signal Ipred is fundamentally bounded by how much
+information Ipast it has about the past signal (panel I), which in turn is limited by the resources necessary to build and operate
+the biochemical network (panel II) [6].
+integration kernel k(t). Since our model obeys Gaussian
+statistics, we use the Gaussian IBM to derive the optimal
+kernel kopt(t) and the information bound, defined to be
+the maximum predictive information as set by the past
+information [12] (see Appendix C).
+Markovian signals
+Optimal prediction of Markovian signals: biochemical
+copying
+Arguably the most elementary type of signal, albeit
+perhaps the hardest to predict, is a Markovian signal.
+We consider a Markovian signal ℓ(t), of which the devia-
+tions δℓ(t) = ℓ(t) − ¯ℓ from its mean ¯ℓ follow an Ornstein-
+Uhlenbeck (OU) process:
+δ ˙ℓ = −δℓ(t)/τℓ + ηℓ(t),
+(3)
+where τℓ is the correlation time of the fluctuations, and
+ηℓ(t) is Gaussian white noise, ⟨η(t)η(t′)⟩ = 2σ2
+ℓ/τℓ δ(t −
+t′), with σ2
+ℓ the amplitude of the signal fluctuations. This
+input signal obeys Gaussian statistics, characterized by
+⟨δℓ(0)δℓ(t)⟩ = σ2
+ℓ exp(−t/τℓ). The optimal mapping is
+therefore a linear one. Utilizing the Gaussian IBM frame-
+work [12], we find that the optimal integration kernel is
+given by (see Appendix C 2)
+kopt(t − t′) = aδ(t − t′).
+(4)
+This optimal integration kernel corresponds to a signaling
+system that copies the current input into the output.
+This is intuitive, since for a Markovian signal there is
+no additional information in the past signal that is not
+already contained in the present one. The prefactor a
+determines the gain ∂¯x/∂¯ℓ, which together with the noise
+strength σ2
+ηx (Eq. 1) and the signal amplitude σ2
+ℓ set the
+magnitude of the past and predictive information, Ipast
+and Ipred, respectively (see Appendix C 1).
+Fig. 2-I shows the maximum predictive information as
+set by the past information. This information bound ap-
+plies to any linear system that needs to predict a Marko-
+vian signal. How close can biochemical systems come to
+this bound?
+Push-pull network can be at the information bound, yet
+increase the predictive and past information by moving away
+from it
+Although the upper bound on the accuracy of predic-
+tion is determined by the signal statistics, how close cells
+can come to this bound depends on the topology of the
+cellular signaling system, and the resources devoted to
+building and operating it. A network motif that could
+reach the information bound for Markovian signals is the
+push-pull network (Fig. 2), because it is at heart a copy-
+ing device: it samples the input by copying the state of
+
+4
+I
+II
+predictive information (bits)
+resources 
+past information (bits)
+FIG. 2. The optimal push-pull network is not at the
+information bound. Panel I: The black line is the informa-
+tion bound that maximizes the predictive information Ipred =
+I(x0; ℓτ) for a given past information Ipast = I(x0; Lp). The
+red curve shows Ipred against Ipast for systems in which Ipred
+has been maximized for a given resource cost C = RT + XT.
+The blue curve shows Ipred versus Ipast for systems where
+Ipast has been maximized for a given C. Panel II shows Ipast
+against C for the corresponding systems.
+The forecast in-
+terval is τ = τℓ. The optimization parameters are the ratio
+XT/RT, τr, p and f (see Appendix E). Parameter values:
+(σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.
+the input, e.g. the ligand-binding state of a receptor or
+the activation state of a kinase, into the activation state
+of the output, e.g. phosphorylation state of the readout
+[10, 11, 23].
+We model the push-pull network in the linear-noise
+approximation:
+δ ˙
+RL = bδℓ(t) − δRL(t)/τc + ηRL(t),
+(5)
+˙
+δx∗ = γ δRL(t) − δx∗(t)/τr + ηx(t).
+(6)
+Here, δRL represents the number of ligand-bound recep-
+tors and δx∗ the number of modified readout molecules,
+defined as deviations from their mean values; b and γ
+are parameters that depend on the number of recep-
+tor and readout molecules, RT and XT respectively, the
+fraction of ligand-bound receptors p and active readout
+molecules f; ηRL and ηx are Gaussian white noise terms
+(see Appendix E). Key parameters are the correlation
+time of receptor-ligand binding, τc, and the relaxation
+time of x∗, τr. The latter determines for how long x∗
+carries information on the ligand-binding state of the re-
+ceptor and thus sets the integration time. The readout-
+modification dynamics yield an exponential integration
+kernel k(t) ∝ exp(−t/τr), which in the limit τr → 0 re-
+duces to a δ-function, hinting that the system may reach
+the information bound.
+How much information cells can extract from the past
+signal depends on the resources devoted to building and
+operating the network (Fig. 2-II). We define the total
+resource cost to be:
+C = λ(RT + XT) + c1XT∆µ/τr
+(7)
+The first term expresses the fact that over the course
+of the cell cycle all components need to be duplicated,
+which means that they have to be synthesized at a speed
+that is at least the growth rate λ. The second term de-
+scribes the chemical power that is necessary to run the
+push-pull network [10, 11]; it depends on the flux through
+the network, XT/τr, and the free-energy drop ∆µ over a
+cycle, e.g. the free energy of ATP hydrolysis in the case
+of a phosphorylation cycle. The coefficient c1 describes
+the relative energetic cost of synthesising the components
+during the cell cycle versus that of running the system.
+For simplicity, we first consider the scenario that the cost
+is dominated by that of protein synthesis, setting c1 → 0.
+While in this scenario RT + XT is constrained, XT/RT
+and other system parameters are free for optimization.
+The available resources put a hard bound on the in-
+formation Ipast that can be extracted from the past sig-
+nal, which in turn sets a hard limit on the predictive
+information Ipred (Fig. 1C). To maximize the predictive
+information, it therefore seems natural to maximize the
+past information Ipast for a given resource cost C. The
+blue line in Fig. 2-II shows the result for the push-pull
+network. We then compute the corresponding predictive
+information for the systems along this line, which is the
+blue line in Fig. 2-I. Strikingly, the resulting information
+curve lies far below the information bound, i.e. the upper
+bound on the predictive information as set by the past
+information (black line, Fig. 2-I). This shows that sys-
+tems that maximize past information under a resource
+constraint, do not in general also maximize predictive in-
+formation. It implies that not all bits of past information
+are equally predictive about the future.
+Precisely because not all bits of past information are
+equally predictive about the future, it is paramount to
+directly maximize the predictive information for a given
+resource cost in order to obtain the most efficient pre-
+diction device. This yields the red lines in panels I and
+II in Fig. 2.
+It can be seen that the predictive infor-
+mation is higher while the past information is lower, as
+compared to the information curves of the systems opti-
+mized for maximizing the past information under a re-
+source constraint (blue lines). It reflects the idea that not
+all bits are equally predictive. More surprisingly, while
+the bound on the predictive information as set by the
+resource cost (red line panel I) is close to the bound on
+the predictive information as set by the past information
+(black line), it does remain lower. This is surprising, be-
+cause the push-pull network is a copying device [10, 23],
+which can, as we will also show below, reach the latter
+bound. These two observations together imply that not
+all bits of past information are equally costly.
+If they
+were, the cell would select under the two constraints the
+same bits based on their predictive information content,
+and the bound on the predictive information as set by
+
+5
+the resource cost would overlap with that as set by the
+past information.
+We thus find that not all bits of past information are
+equally predictive, nor equally costly. As we show next,
+it implies that the optimal information processing system
+faces a trade-off between using those bits of past infor-
+mation that are most informative about the future and
+those that are cheapest.
+Trade-off between cost and predictive power per bit
+To understand the connection between predictive and
+past information, and resource cost, we map out the re-
+gion in the information plane that can be reached given
+a resource constraint C (Fig. 3A, green region). We im-
+mediately make two observations.
+Firstly, the system
+can indeed reach the information bound. Secondly, the
+system can increase both the past and the predictive in-
+formation by moving away from the bound. To elucidate
+these two observations, we investigate the system along
+the isocost line of C = 104, which together with the in-
+formation bound envelopes the accessible region for the
+maximum resource cost C ≤ 104.
+Along the isocost line,
+the ratio of the number
+of
+readout
+over
+receptor
+molecules
+is
+XT/RT
+=
+2
+�
+p/(1 − p)
+�
+1 + τr/τc (see Appendix E 3). This can be
+understood intuitively using the optimal resource alloca-
+tion principle [10]. It states that in a sensing system that
+employs its proteins optimally, the total number of inde-
+pendent concentration measurements at the level of the
+receptor during the integration time τr, RT(1 + τr/τc),
+equals the number of readout molecules XT that store
+these measurements, so that neither the receptors nor
+the readout molecules are in excess. This design prin-
+ciple specifies, for a given integration time τr, the ratio
+XT/RT at which the readout molecules sample each re-
+ceptor molecule roughly once every receptor correlation
+time τc.
+While the optimal allocation principle gives the opti-
+mal ratio XT/RT of the number of readouts over recep-
+tors for a given integration time τr, it does not prescribe
+what the optimal integration time τropt, and hence (glob-
+ally) optimal ratio Xopt
+T /Ropt
+T , is that maximizes Ipred for
+a given resource constraint C = RT +XT. Fig. 3B shows
+that as the distance θ along the isocost line is increased,
+τr and hence XT/RT increase monotonically. Near the
+information bound, corresponding to θ = 0, the integra-
+tion time τr is zero and the number of readout molecules
+equals the number of receptor molecules: XT = RT. In
+this limit, the push-pull network is an instantaneous re-
+sponder, with an integration kernel given by Eq. 4; only
+the finite receptor correlation time τc prevents the sys-
+tem from fully reaching the information bound. Yet, as
+θ increases and the system moves away from the bound,
+the predictive and past information first rise along the
+contour, and thus with XT/RT and τr, before they even-
+tually both fall.
+To understand why the predictive and past informa-
+tion first rise and then fall with XT/RT and τr, we note
+that each readout molecule constitutes 1 physical bit and
+that its binary state (phosphorylated or not) encodes at
+most 1 bit of information on the ligand concentration.
+The number of readout molecules XT thus sets a hard
+upper bound on the sensing precision and hence the pre-
+dictive information. To raise this bound, XT must be
+increased. For a given resource constraint C = RT +XT,
+XT can only be increased if the number of receptors RT
+is simultaneously decreased. However, the cell infers the
+concentration not from the readout molecules directly,
+but via the receptor molecules: a readout molecule is a
+sample of the receptor that provides at most 1 bit of in-
+formation about the ligand-binding state of a receptor
+molecule, which in turn provides at most 1 bit of infor-
+mation about the input signal. To raise the lower bound
+on the predictive information, the information on the in-
+put must increase at both the receptor and the readout
+level.
+To elucidate how this can be achieved, we note that the
+maximum number of independent receptor samples and
+hence concentration measurements is given by N max
+I
+=
+min(XT, RT(1 + τr/τc)) [10]. For θ > 0, the system can
+increase N max
+I
+if, and only if, XT and RT(1 + τr/τc) can
+be raised simultaneously.
+This can be achieved, while
+obeying the constraint C = XT + RT, by decreasing RT
+yet increasing τr (Fig. 3B). This is the mechanism of time
+averaging, which makes it possible to increase the num-
+ber of independent receptor samples [11], and explains
+why both the predictive and the past information initially
+increase (Fig. 3C). However, as τr is raised further, the
+receptor samples become older: the readout molecules in-
+creasingly reflect receptor states in the past that are less
+informative about the future ligand concentration. The
+collected bits of past information have become less pre-
+dictive about the future (Fig. 3C). For a given resource
+cost, the cell thus faces a trade-off between maximizing
+the number of physical bits of past information (i.e. the
+receptor samples XT) and the predictive information per
+bit. This antagonism gives rise to an optimal integration
+time τropt that maximizes the total predictive informa-
+tion Ipred (Fig. 3C).
+Interestingly, while Ipred decreases beyond τropt, the
+past information Ipast first continues to rise because
+N max
+I
+still increases. However, when the integration time
+becomes longer than the input signal correlation time,
+the correlation between input and output will be lost
+and Ipast will fall too.
+Chemical power prevents the system from reaching the
+information bound
+So far, we have only considered the cost of maintain-
+ing the cellular system, the protein cost C = RT + XT.
+Yet, running a push-pull network also requires energy.
+As Eq. 7 shows, the running cost scales with the flux
+
+6
+A
+B
+C
+FIG. 3. The push-pull network maximizes the predictive power under a resource constraint by moving away
+from the information bound. (A) The region of accessible predictive information Ipred = I(x0; ℓτ) and past information
+Ipast = I(x0; Lp) in the push-pull network under a resource constraint C ≤ (RT + XT), for the Markovian signals specified by
+Eq. 3 (green). The black line is the information bound at which Ipred is maximized for a given Ipast. The push-pull network
+can be at the information bound (black points), but maximizing Ipred for a resource constraint C moves the system away from
+it. The red and blue lines connect, respectively, the points where Ipred and Ipast are maximized along the green isocost lines
+(the contourlines of constant C); they correspond to the red and blue lines in Fig. 2, respectively. The accessible region of
+Ipred and Ipast for a given C has been obtained by optimizing over τr, p, f, and XT/RT. The forecast interval is τ = τℓ.
+(B) The integration time τr over the receptor correlation time τc, τr/τc, and the ratio of the number of readout and receptor
+molecules, XT/RT, as a function of the distance θ along the isocost line corresponding to C = 104 in panel A; the red and
+blue points denote where Ipred and Ipast are maximized along the contourline, respectively. For θ → 0, τr → 0: the system is
+an instantaneous responder, which is essentially at the information boundary; as predicted by the optimal resource allocation
+principle, XT = RT. The system can increase Ipred and Ipast by increasing τr and XT/RT. (C) While this decreases the
+predictive information Ipred per physical bit of past information, Ipred/XT (dashed line), increasing XT/RT does increase the
+number of physical bits per resource cost, XT/C (purple line). This trade-off gives rise to an optimal predictive information
+per resource cost, Ipred/C (red dot on solid black line). Parameter values unless specified: (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.
+around the phosphorylation cycle, which is proportional
+to the inverse of the integration time, τr−1. The power
+thus diverges for τr → 0. Since the information bound is
+reached precisely in this limit, it is clear that the chem-
+ical power prevents the push-pull network from reaching
+the bound (see Fig. 7 in the appendix).
+Non-Markovian signals
+Predicting the future change
+The push-pull network can optimally predict Marko-
+vian signals, yet not all signals are expected to be Marko-
+vian. Especially organisms that navigate through an en-
+vironment with directional persistence will sense a non-
+Markovian signal, as generated by their own motion.
+Moreover, when these organisms need to climb a con-
+centration gradient, as E. coli during chemotaxis, then
+knowing the change in the concentration is arguably more
+useful than knowing the concentration itself. Indeed, it
+is well known that the kernel of the E. coli chemotaxis
+system detects the (relative) change in the ligand con-
+centration by taking a temporal derivative of the concen-
+tration [15]. However, as we will show here, the converse
+statement is more subtle. If the system needs to predict
+the (future) change in the signal, then the optimal ker-
+nel is not necessarily one that is based on the derivative
+only: in general, the optimal kernel uses a combination of
+the signal value and its derivative. However, the E. coli
+chemotaxis system can respond to concentrations that
+vary between the dissociation constants of the inactive
+and active state of the receptors, which differ by several
+orders of magnitude [24]. This range of possible back-
+ground concentrations is much larger than the typical
+concentration change over the orientational correlation
+time of the bacterium.
+As our analysis below reveals,
+in this regime the optimal kernel is a perfectly adaptive,
+derivative-taking kernel that is insensitive to the current
+signal value, precisely like that of the E. coli chemotaxis
+system [15, 25–28]. Our analysis thus predicts that this
+system has an adaptive kernel, because this is the opti-
+mal kernel for predicting concentration derivatives over
+a broad range of background concentrations.
+To reveal the signal characteristics that control the
+shape of the optimal integration kernel, we will consider
+the family of signals that are generated by a harmonic
+
+7
+oscillator:
+δ ˙ℓ = v(t),
+(8)
+˙v = −ω2
+0δℓ(t) − v(t)/τv + ηv(t),
+(9)
+where δℓ is the deviation of ligand concentration from its
+mean ¯ℓ, v its derivative, τv a relaxation time, ηv a Gaus-
+sian white noise term, and the frequency ω2
+0 = σ2
+v/σ2
+ℓ
+controls the variance σ2
+ℓ of the concentration and that of
+its derivative σ2
+v.
+Using the IBM framework it can be shown that the
+optimal encoding that allows the system to reach the
+information bound, is based on a linear combination of
+the current concentration ℓ(t) and its derivative v(t), such
+that the output x(t) is given by (Appendix C 3):
+x(t) = aδℓ(t)
+σℓ
++ bv(t)
+σv
++ ηx(t).
+(10)
+This can be understood by noting that while the signal
+of Eqs. 8 and 9 is non-Markovian in the space of ℓ, it is
+Markovian in ℓ and v: all the information on the future
+signal is thus contained in the current concentration and
+its derivative. To maximize the predictive information
+Ipred = I(x0; vτ) between the current output x0 and the
+future derivative of the input vτ for a given amount of
+past information Ipast = I(x0; Lp), i.e to reach the infor-
+mation bound for predicting the future signal derivative,
+the coefficients must obey
+aopt = G⟨δℓ(0)δv(τ)⟩
+σℓσv
+≡ Gρℓ0vτ ,
+(11)
+bopt = G⟨δv(0)δv(τ)⟩
+σ2v
+≡ Gρv0vτ .
+(12)
+Here, G is the gain, which together with the noise σ2
+ηx sets
+the scale of Ipred and Ipast, ρℓ0vτ is the cross-correlation
+coefficient between the current concentration value ℓ0 and
+the future concentration derivative vτ and ρv0vτ that be-
+tween the current and future derivative (Appendix C 3).
+These expressions can be understood intuitively: if the
+future signal derivative that needs to be predicted is cor-
+related with the current signal derivative, it is useful
+to include in the prediction strategy the current signal
+derivative, leading to a non-zero value of bopt. Perhaps
+more surprisingly, if the future signal derivative is also
+correlated with the current signal value, then the system
+can enhance the prediction accuracy by also including the
+current signal value, yielding a non-zero aopt. Clearly,
+in general, to optimally predict the future signal change,
+the system should base its prediction on both the current
+signal value and its derivative.
+The degree to which the systems bases its prediction on
+the current value versus the current derivative depends
+on the relative magnitudes of aopt and bopt, respectively.
+In Appendix B 2, we show that when the concentration
+change over the timescale τv, σvτv, is much smaller than
+the range of possible concentrations σℓ that the bac-
+terium can experience, i.e. when σvτv ≪ σℓ such that
+ω0 ≪ τ −1
+v , the cross-correlation coefficient ρℓ0vτ vanishes,
+such that aopt becomes zero (see Eq. 11). The optimal
+kernel has become a perfectly adaptive, derivative-taking
+kernel. We emphasize that while we have derived this re-
+sult for the class of signals defined by Eqs. 8 and 9, the
+idea is far more generic. In particular, while we do not
+know the temporal structure of the ligand statistics that
+E. coli experiences, we do know that it can detect con-
+centration changes over a range of background concentra-
+tions that is much wider that the typical concentration
+change over a run, such that the correlation between the
+concentration value and its future change is likely to be
+very small. As our analysis shows, a perfectively adap-
+tive kernel then emerges naturally from the requirement
+to predict the future concentration change.
+While the class of signals specified by Eqs. 8 and 9 is
+arguably limited, it does describe the biologically impor-
+tant regime of chemotaxis in shallow gradients. In the
+limit that ω0 ≪ τv−1, Eq. 9 reduces to ˙v = −v/τv + ηv.
+In shallow gradients, the stimulus only weakly affects
+the swimming behavior, such that the perceived signal
+is mostly determined by the intrinsic orientational dy-
+namics of the bacterium in the absence of a gradient. In
+this regime, the temporal statistics of the concentration
+derivative v is completely determined by the steepness of
+the concentration gradient g and the swimming statistics
+of the bacterium in the absence of a gradient:
+⟨δv(0)δv(τ)⟩ = g2¯ℓ2⟨δvx(0)δvx(τ)⟩ ≃ σ2
+vxe−τ/τvx ,
+(13)
+where the latter is the autocorrelation function of the
+(positional) velocity of the bacterium in the absence of a
+gradient. It is a characteristic of the bacterium, not of
+the environment, and has been measured to decay expo-
+nentially with a correlation time τvx [18], precisely as our
+model, with τv = τvx, predicts. This correlation time is
+on the order of the typical run time of the bacterium in
+the absence of a gradient, τv ∼ 0.9s [18].
+Finite resources prevent the chemotaxis system from taking
+an instantaneous derivative and reaching the information
+bound
+The above analysis indicates that the chemotaxis sys-
+tem seems ideally designed to predict the future concen-
+tration change, because its integration kernel is nearly
+perfectly adaptive [15, 25–28]. But how close can this
+system come to the information bound for the non-
+Markovian signals specified by Eqs. 8 and 9?
+To address this, we consider a molecular model that
+can accurately describe the response of the chemotaxis
+system to a wide range of time-varying signals [29–32].
+In this model, the receptors are partitioned into clusters.
+Each cluster is described via a Monod-Wyman-Changeux
+model [33]. While each receptor can switch between an
+active and an inactive conformational state, the energetic
+cost of having different conformations in the same cluster
+is prohibitively large. Each cluster is thus either active or
+
+8
+inactive. Ligand binding favors the inactive state while
+methylation does the opposite.
+Lastly, active receptor
+clusters can via the associated kinase CheA phosphory-
+late the downstream messenger protein CheY.
+Linearizing around the steady state, we obtain:
+δai(t) = αδmi(t) − βδℓ(t),
+(14)
+δ ˙mi = −δai(t)/(ατm) + ηmi(t),
+(15)
+δ ˙x∗ = γ
+RT
+�
+i=1
+δai(t) − δx∗(t)/τr + ηx(t).
+(16)
+Here, δai(t) and δmi(t) are the deviations of the activ-
+ity and methylation level of receptor cluster i from their
+steady-state values, and RT is the total number of recep-
+tor clusters; δℓ(t) and δx∗(t) are, respectively, the devi-
+ations of the ligand and CheYp concentration from their
+steady-state values; τm and τr are the timescales of re-
+ceptor methylation and CheYp dephosphorylation; ηmi
+and ηx are independent Gaussian white noise sources. In
+Eq. 14, we have assumed that ligand binding is much
+faster than the other timescales in the system, so that
+it can be integrated out. There is therefore no need to
+time average receptor-ligand binding noise, which means
+that, in the absence of running costs, the optimal re-
+ceptor integration time τr is zero. In what follows, we
+set τr to the value measured experimentally, τr ≈ 100ms
+[10, 34]. We consider the non-Markovian signals speci-
+fied by Eqs. 8 and 9 in the physiologically relevant limit
+ω0 → 0, such that the optimal kernel is perfectly adap-
+tive, like that of E. coli. For these signals, we determine
+the accessible region of Ipast and Ipred under a resource
+constraint C = RT + XT (see Fig. 4) by optimizing over
+the methylation time τm and the ratio of readout over
+receptor molecules XT/RT.
+The forecast interval τ is
+set to τv, but we emphasize that the optimal design is
+independent of the value of τ (see Appendix F 4).
+Fig. 4A shows that the chemotaxis system is, in gen-
+eral, not at the information bound that maximizes the
+predictive information Ipred = I(x0; vτ) for a given past
+information Ipast = I(x0; Lp). The optimal systems that
+maximize Ipred under a resource constraint C, marked by
+the red dots, are indeed markedly away from the infor-
+mation bound. Yet, as the resource constraint is relaxed
+and C is increased, the optimal system moves towards
+the bound.
+Panel B shows that the methylation time
+τm rises along the three respective isocost lines of panel
+A. It highlights that there exists an optimal methyla-
+tion time τ opt
+m
+that maximizes the predictive information
+Ipred. Moreover, τ opt
+m
+decreases as the resource constraint
+is relaxed.
+Along the respective isocost lines, XT/RT
+varies only mildly (see Fig. 9 in the appendix).
+These observations can be understood by noting that
+the system faces a trade-off between taking a derivative
+that is recent versus one that is robust. All the infor-
+mation on the future derivative, which the cell aims to
+predict, is contained in the current derivative of the sig-
+nal; measuring the current derivative would allow the
+system to reach the information bound. However, com-
+puting the recent derivative is extremely costly. The cell
+takes the temporal derivative of the ligand concentration
+at the level of the receptor via two antagonistic reac-
+tions that occur on two distinct timescales: ligand bind-
+ing rapidly deactivates the receptor, while methylation
+slowly reactivates it [30]. The receptor ligand-occupancy
+thus encodes the current concentration, the methylation
+level stores the average concentration over the past τm,
+and the receptor activity reflects the difference between
+the two—the temporal derivative of the signal over the
+timescale τm. To obtain an instantaneous derivative, τm
+must go to zero. However, this dramatically reduces the
+gain; in fact, in this limit, the gain is zero, because the
+receptor activity instantly adapts to the change in the
+ligand concentration. Since the push-pull network down-
+stream of the receptor is a device that samples the re-
+ceptor stochastically [10, 36], the gain, i.e. the change in
+the receptor activity due to the signal, must be raised to
+lift the signal above the sampling noise. This requires a
+finite methylation time τm: as we show in Appendix F 3,
+the gain increases monotonically with τm. The trade-off
+between a recent derivative and a reliable one gives rise
+to an optimal methylation time τ opt
+m
+that maximizes the
+predictive information for a given resource cost.
+The same analysis also explains why the optimal
+methylation time τ opt
+m
+decreases and the predictive infor-
+mation increases when the resource constraint is relaxed.
+The sampling noise in estimating the average receptor
+activity decreases as the number of readout molecules
+increases [10, 36]. A smaller gain is thus required to lift
+the signal above the sampling noise. In addition, a larger
+number of receptors decreases the noise in the methyla-
+tion level, which also allows for a smaller gain, and hence
+a smaller methylation time. These two effects together
+explain why τ opt
+m
+decreases and Ipred increases with C.
+Fig. 4A also shows that the past information Ipast =
+I(x0; Lp) does not return to zero along the contourline of
+constant resource cost. Along the contourline, the methy-
+lation time τm rises (Fig. 4B). While the predictive infor-
+mation Ipred exhibits an optimal methylation time τmopt,
+the past information Ipast continues to rise with τm be-
+cause the system increasingly becomes a copying device,
+rather than one that takes a temporal derivative.
+Comparison with experiment
+To test our theory, we study the predictive power of
+the E. coli chemotaxis system as a function of the steep-
+ness of the ligand concentration gradient, keeping the
+resource constraint at the biologically relevant value of
+C = RT + XT = 104 [35]. Panel C of Fig. 4 shows Ipred
+and Ipast for cells swimming in an exponential concen-
+tration gradient ℓ(x) = ℓ0egx, for different values of the
+gradient steepness g; along the green iso-steepness lines
+τm is varied and XT/RT is optimized to maximize Ipred
+and Ipast, with the red dots marking τ opt
+m , while along
+
+9
+A
+B
+C
+FIG. 4. Finite resources prevent chemotaxis system from reaching the information bound. (A) The region of
+accessible predictive information Ipred = I(x0; vτ) and past information Ipast = I(x; Lp) for the chemotaxis system under a
+resource constraint C = RT + XT, for the non-Markovian signals specified by Eqs. 8 and 9 (green). The black line shows the
+information bound at which Ipred is maximized for a given Ipast. The chemotaxis system is not at the information bound, but
+it does move towards it as C is increased. The red line connects the red points where Ipred is maximized for a given resource
+cost C. The accessible region of Ipred and Ipast under a given resource constraint C = RT + XT is obtained by optimizing over
+the methylation time τm and the ratio of readout over receptor molecules XT/RT. The forecast interval is τ = τv. (B) The
+methylation time τm over the input correlation time τv as a function of the distance θ along the three respective isocost lines
+shown in panel A. The methylation time τm increases along the isocost line, but there exists an optimal τm that maximizes
+the predictive information, marked by the red points; θ → 0 corresponds to the origin of panel A, (Ipred, Ipast) = (0, 0); the
+points where θ = 0.2 along the isocost lines of panel A are marked with a bar. As the resource constraint is relaxed (higher
+C), the optimal τm decreases: the system moves towards the information bound, where it takes an instantaneous derivative,
+corresponding to τr, τm → 0. (C) The contourlines of Ipred and Ipast for increasing values of the steepness g of an exponential
+ligand concentration gradient ℓ(x) = ℓ0egx, keeping the total resource cost fixed at C = RT + XT = 104; τm and XT/RT
+have been optimized. It is seen that the maximal predictive information Ipred under the resource constraint C (marked by
+the red points) increases with the gradient steepness. The blue line shows Ipred and Ipast for the E. coli chemotaxis system
+with τm = 10s and XT = RT = 5000 fixed at their measured values [35]. Our analysis predicts that this system has been
+optimized to detect shallow gradients. Parameter values unless specified: τr = 100ms [10, 34]; τv = 0.9s and σ2
+v = g2¯ℓ2σ2
+vx,
+with ¯ℓ = 100µM and σ2
+vx = 157.1µm2s−2 [18]; ω0 → 0; g is given in units of mm−1; in A, g = 4/mm.
+the blue line τm and XT and RT are fixed at their exper-
+imentally measured values [29, 30, 35]. Clearly, both the
+predictive and the past information rise as the gradient
+steepness g increases—a steeper concentration gradient
+yields a larger change in the concentration, and thus a
+stronger signal.
+More interestingly, in the optimal system Ipred rises
+much faster with Ipast (red line) than in the E. coli system
+(blue line). A steeper gradient g yields a stronger input
+signal, which raises the signal above the sampling noise
+more. This allows the optimal system to take a more re-
+cent derivative, with a smaller τm, which is more informa-
+tive about the future. In contrast, the methylation time
+τm of the E. coli chemotaxis system is fixed. As Fig. 4C
+shows, this value is beneficial for detecting shallow gra-
+dients, g ≲ 0.2mm−1. Moreover, in this regime, not only
+Ipred but also Ipast are close to the respective values for
+the optimal system. For steeper gradients Ipast becomes
+much higher in the E. coli system than in the optimal
+one, even though Ipred remains lower.
+The bacterium
+increasingly collects information that is less informative
+about the future. Taken together, these results strongly
+suggest that the system has been optimized to predict
+future concentration changes in shallow gradients, which
+necessitate a relatively long methylation time.
+DISCUSSION
+Cellular systems need to predict the future signal by
+capitalizing on information that is contained in the past
+signal. To this end, they need to encode the past sig-
+nal into the dynamics of the intracellular biochemical
+network from which the future input is inferred.
+To
+maximize the predictive information for a given amount
+of information that is extracted, the cell should store
+those signal characteristics that are most informative
+about the future signal. For a Markovian signal obeying
+an Ornstein-Uhlenbeck process this is the current signal
+value, while for the non-Markovian signal corresponding
+to an underdamped particle in a harmonic well, this is
+the current signal value and its derivative. As we have
+seen here, cellular systems are able to extract these sig-
+
+10
+nal characteristics: the push-pull network can copy the
+current input into the output, while the chemotaxis net-
+work can take an instantaneous derivative. We have thus
+demonstrated that at least for two classes of signals, cel-
+lular systems are in principle able to extract the most
+predictive information, allowing them to reach the infor-
+mation bound.
+Yet, our analysis also shows that extracting the most
+relevant information can be exceedingly costly. To copy
+the most recent input signal into the output, the integra-
+tion time of the push-pull network needs to go to zero,
+which means that the chemical power diverges. More-
+over, taking an instantaneous derivative reduces the gain
+to zero, such that the signal is no longer lifted above
+the inevitable intrinsic biochemical noise of the signalling
+system. In fact, taking the chemical power cost to drive
+the adaptation cycle into account [27, 37] would push the
+system away from the information bound even more.
+While information is a resource—the cell cannot pre-
+dict the future without extracting information from the
+past signal—the principal resources that have a direct
+cost are time, building blocks and energy. The predic-
+tive information per protein and energy cost is therefore
+most likely a more relevant fitness measure than the pre-
+dictive information per past information. Our analysis
+reveals that, in general, it is not optimal to operate at
+the information bound: cells can increase the predictive
+information for a given resource constraint by moving
+away from the bound. Increasing the integration time in
+the push-pull network reduces the chemical power and
+makes it possible to take more concentration measure-
+ments per protein copy. And increasing the methylation
+time in the chemotaxis system increases the gain. Both
+enable the system to extract more information from the
+past signal. Yet, increasing the integration time or the
+methylation time also means that the information that
+has been collected, is less informative about the future
+signal. This interplay gives rise to an optimal integration
+and methylation time, which maximize the predictive in-
+formation for a given resource constraint. This argument
+also explains why the respective systems move towards
+the information bound when the resource constraint is
+relaxed: Increasing the number of receptor and readout
+molecules allows the system to take more instantaneous
+concentration measurements, which makes time averag-
+ing less important, thus reducing the integration time.
+Increasing the number of readout molecules also reduces
+the error in sampling the receptor state. This makes it
+easier to detect a change in the receptor activity result-
+ing from the signal, thus allowing for a smaller dynamical
+gain and a shorter methylation time.
+Information theory shows that the amount of transmit-
+ted information depends not only on the characteristics
+of the information processing system, but also on the
+statistics of the input signal. While much progress has
+been made in characterizing cellular signalling systems,
+the statistics of the input signal is typically not known,
+with a few notable exceptions [38].
+Here, we have fo-
+cussed on two classes of input signals, but it seems likely
+that the signals encountered by natural systems are much
+more diverse. It will be interesting to extend our analy-
+sis to signals with a richer temporal structure [9], and see
+whether cellular systems exist that can optimally encode
+these signals for prediction.
+Finally, while we have analyzed the design of cellular
+signaling networks to optimally predict future signals, we
+have not addressed the utility of information for function
+or behavior. It is clear that many functional or behavioral
+tasks, like chemotaxis [18], require information, but what
+the relevant bits of information are is poorly understood
+[7]. Moreover, cells ultimately employ their resources—
+protein copies, time, and energy—for function or behav-
+ior, not for processing information per se. Here, we have
+shown that maximizing predictive information under a
+resource constraint, C → Ipast → Ipred, does not nec-
+essarily imply maximizing past information. This hints
+that optimizing a functional or behavioral task under a
+resource constraint, C → Ipred → function, may not im-
+ply maximizing the predictive information necessary to
+carry out this task.
+ACKNOWLEDGMENTS
+We thank Jenny Poulton, Manuel Reinhardt, Michael
+Vennettilli and Daan de Groot for many useful discus-
+sions. This work is part of the Dutch Research Coun-
+cil (NWO) and was performed at the research institute
+AMOLF. This project has received funding from the
+European Research Council (ERC) under the European
+Union’s Horizon 2020 research and innovation program
+(grant agreement No. 885065).
+
+11
+Appendix A: General
+1.
+Linear signalling networks
+Since the systems studied in the main text have a single steady state, we will study them in the linear-noise
+approximation [39]. For non-linear systems, the quality of the approximation improves with system size, but it can
+already be remarkably good for systems with only 10 copies [20, 22, 40]. In the linear-noise approximation, we expand
+the rate equations to first order around the steady state of the mean-field chemical rate equations, and compute the
+noise at this steady state. In this approximation the network dynamics are a multidimensional Ornstein-Uhlenbeck
+(OU-)process:
+˙δy = Gδs(t) + J δy(t) + Bξ(t),
+(A1)
+where δs(t) is a length k vector of input signals and δy is the vector of all network species of length n, both
+defined in terms of deviations from their mean. The vector ξ(t) describes the m independent white noise processes
+associated with the m network reactions; they have zero mean, unit variance, and are delta correlated: ⟨ξi(t)⟩ = 0,
+⟨ξi(t)ξj(t′)⟩ = δijδ(t − t′), with δij the Kronecker delta. The n × n matrix J is the Jacobian of the network, the n × k
+signal gain matrix G describes the strength by which each signal impacts each species directly, the n × m matrix B
+contains the noise strengths. The eigenvalues of the Jacobian J must be negative for the system to be stable, and we
+require all signals to be stationary.
+2.
+Integration kernels, power spectra, and correlation functions
+We continue by deriving the stationary auto-correlation matrix of a multidimensional OU-process, such as Eq. A1,
+via the networks’ power spectra. The power spectrum of a real-valued random process X(t) is the squared modulus
+of its Fourier transform: Sx(ω) = ⟨δ˜x(−ω)δ˜x(ω)⟩ and Sx→y(ω) = ⟨δ˜x(−ω)δ˜y(ω)⟩. Throughout this work we use
+the following conventions for the Fourier transform and its inverse: F{f(t)} ≡ ˜f(ω) =
+� ∞
+−∞ dtf(t) exp(−iωt) and
+F−1{ ˜f(ω)} = 1/(2π)
+� ∞
+−∞ dω ˜f(ω) exp(iωt) = f(t). To obtain the correlation functions from the power spectra we
+invoke the Wiener-Khinchin theorem.
+The general solution to Eq. A1 is
+δy(t) =
+� t
+−∞
+dt′ eJ (t−t′) (Gδs(t′) + Bξ(t′)) ,
+(A2)
+which shows the two contributions to the time dependent solution: that of the external signal and that of the internal
+noise. The n × k matrix eJ (t−t′)G contains the integration kernels, its (i, j)th entry determines how the jth signal
+affects the ith system component over time. The n × m matrix eJ (t−t′)B is similar, but contains the functions that
+map the noise terms onto the system components. These matrices can be obtained by taking the Fourier transform
+of Eq. A1 and solving for δ˜y(ω)
+iωδ˜y(ω) = Gδ˜s(ω) + J δ˜y(ω) + B˜ξ(ω),
+(A3)
+δ˜y(ω) = (iωIn − J )−1 �
+Gδ˜s(ω) + B˜ξ(ω)
+�
+.
+(A4)
+Using the convolution theorem to take the Fourier transform of Eq. A2, and comparing the result to Eq. A4, now
+shows that F{eJ (t−t′)} = (iωIn − J )−1. We obtain for the power-spectra of the network components
+Sy(ω) = ⟨δ ˜y(−ω)δ ˜y(ω)T ⟩,
+= G(−ω)Ss(ω)G(ω)T + |N(ω)|2,
+(A5)
+with the matrices of frequency dependent gains G(ω) ≡ (iωIn − J )−1G, and frequency dependent noise N(ω) ≡
+(iωIn−J )−1B. The cross terms vanish because the fluctuations of the external signal are uncorrelated from the internal
+noise. Furthermore, the power spectrum of a white noise process is constant, and all the noise terms are independent
+of one another, such that the spectral density of the noise vector is the identity matrix ⟨˜ξ(−ω)˜ξ(ω)T ⟩ = Im. We
+also need to consider the cross-spectra between the signals and the network components, specifically we will need the
+spectra from the network to the signals
+Sy→s(ω) = ⟨δ ˜y(−ω)δ˜s(ω)T ⟩,
+= G(−ω)Ss(ω).
+(A6)
+
+12
+From Eq. A5 and Eq. A6 we can obtain all necessary correlation functions and (co-)variances, by taking the inverse
+Fourier transform of the component of interest (for a variance we can directly set t = 0). The advantage of using
+this form, is that the contribution of each signal and of the noise terms appear separately. When we are for example
+interested in a variance that is only caused by noise, we can omit the terms depending on the signal power spectra,
+and vice versa. Moreover, the power spectra are usually simpler in form than the corresponding correlation functions.
+The covariance and auto-correlation matrices can also be found by solving Eq. A2 directly in the time domain; the
+solutions are shown here for completeness. For a derivation, see for example the work by Vennettilli et al. [41]. In this
+case it is most convenient to include the signals as system components, we thus have a new Jacobian J ′ and a new
+noise strength matrix B′ which include all network components and the signals themselves. The covariance matrix C
+is then obtained by solving the Lyapunov equation
+J ′C + CJ ′T + B′B′T = 0,
+(A7)
+and the correlation matrix is given by
+C(τ) = eJ ′τC
+for τ > 0.
+(A8)
+Appendix B: Signals and statistics
+1.
+Markovian signal
+For the Markovian ligand concentration dynamics we use a 1-dimensional OU-process
+δ ˙ℓ = −δℓ/τℓ + ηℓ(t),
+(B1)
+where the ligand concentration is defined in terms of the deviation from its mean δℓ = ℓ(t) − ¯ℓ. The correlation
+time is give by τℓ, and the noise ηℓ(t) is derived from a unit white noise process ηℓ(t) ≡ σℓ
+�
+2/τℓξ(t), such that
+⟨ηℓ(t)ηℓ(t′)⟩ = 2σ2
+ℓ/τℓδ(t − t′). We obtain for the steady-state auto-correlation using Eq. A7 and Eq. A8:
+⟨δℓ(τ)δℓ(0)⟩ = σ2
+ℓe−τ/τℓ.
+(B2)
+2.
+Non-Markovian signal
+Not all ligand concentration trajectories encountered by cells are expected to be Markovian. For example, E. coli
+swims in its environment with a speed which exhibits persistence. This leads to an auto-correlation function for the
+concentrations’ derivative which does not decay instantaneously [18]. To model such a persistent signal, we use the
+classical model of a particle in a harmonic well
+δ ˙ℓ = v(t),
+˙v = −ω2
+0δℓ(t) − v(t)/τv + ηv(t),
+(B3)
+where ω0 =
+�
+k/m, with k the spring constant and m the mass of the particle, τv is a relaxation timescale, and
+ηv(t) = σv
+�
+2/τvξ(t), with ξ(t), as used throughout, a Gaussian white noise process of unit variance, and σv the
+standard deviation of v. If the signal would obey the fluctuation-dissipation relation, then mσ2
+v = kBT, but since the
+biochemical signal could very well be generated via an active process this relation may not hold. This process can be
+expressed as a 2-dimensional OU-process with:
+J =
+�
+0
+1
+−ω2
+0
+−1/τv
+�
+,
+(B4)
+B =
+�0
+0
+0
+σv
+�
+2/τv
+�
+.
+(B5)
+We find for the covariance matrix, using Eq. A7:
+C =
+�
+σ2
+ℓ
+σℓv
+σℓv
+σ2
+v
+�
+= σ2
+v
+�
+1/ω2
+0
+0
+0
+1
+�
+.
+(B6)
+
+13
+Using Eq. A8 we obtain the auto-correlation matrix in the overdamped regime, τ −1
+v
+> 2ω0,
+C(τ) =
+�⟨δℓ(τ)δℓ(0)⟩
+⟨δℓ(τ)δv(0)⟩
+⟨δv(τ)δℓ(0)⟩
+⟨δv(τ)δv(0)⟩
+�
+,
+=
+�
+�
+�
+σ2
+ℓe−µτ/2 �
+cosh(ρτ) + µ
+2ρ sinh(ρτ)
+�
+σ2
+ve−µτ/2 1
+ρ sinh(ρτ)
+−σ2
+ve−µτ/2 1
+ρ sinh(ρτ)
+σ2
+ve−µτ/2 �
+cosh(ρτ) − µ
+2ρ sinh(ρτ)
+�
+�
+�
+� ,
+(B7)
+where ρ =
+�
+µ2/4 − ω2
+0, with µ = τv−1. The range of ligand concentrations which E. coli might encounter is very
+large, based on the dissociation constants of the inactive and active receptor conformations, which for the Tar-MeAsp
+receptor ligand combination respectively are KI
+D = 18µM and KA
+D = 2900µM [42, 43]. This suggests that the variance
+in the ligand concentration is very large relative to that of the derivative of the ligand concentration, which is set by
+the swimming behaviour of the cell. For this reason we specifically focus on the limit where ω0 → 0, which corresponds
+to a vanishingly small spring constant, or a harmonic potential which becomes extremely wide. The variance in the
+concentration σ2
+ℓ then diverges, the normalized correlation functions in this limit are
+lim
+ω0→0
+�
+�
+⟨δℓ(τ)δℓ(0)⟩
+σ2
+ℓ
+⟨δℓ(τ)δv(0)⟩
+σℓσv
+⟨δv(τ)δℓ(0)⟩
+σℓσv
+⟨δv(τ)δv(0)⟩
+σ2v
+�
+� =
+�1
+0
+0
+e−µτ
+�
+.
+(B8)
+Appendix C: Information bottleneck framework and solutions
+Anticipating future environmental conditions allows for timely adaptation.
+However, storing information costs
+resources such as proteins, energy and time, and not all information in the past ligand concentrations will be relevant
+for predicting the signal’s future state. Assuming that resources are in limited supply, this means that cells must be
+efficient in which, and how much information they store. This is elegantly captured in the Information Bottleneck
+Method (IBM), which describes the problem of maximizing the information on the future signal while minimizing the
+information on the past signal that is stored in the network output, from which the future input is predicted [8]. The
+objective function for the prediction of a variable of interest zτ ≡ z(t + τ) is:
+max
+P (X0|Lp) :
+L = I(x0; zτ) − γI(x0; Lp).
+(C1)
+The value of the sensing system output at the current time t is x0 ≡ x(t). The variable of interest zτ at a future
+time t + τ is the future concentration ℓτ ≡ ℓ(t + τ) for the Markovian signal, and the future concentration derivative
+vτ ≡ v(t+τ) for the non-Markovian signal. Since the system of interest needs to predict one signal characteristic (either
+the future signal value or its derivative), one output component is sufficient for encoding the required information,
+as we describe in more detail below. The vector Lp = (δℓ(0), δℓ(−∆t), . . . , δℓ(−(N − 1)∆t))T is the past trajectory
+of ligand concentrations of length N, discretized with timestep ∆t. The mutual information between the current
+system output and the future property of interest is the predictive information Ipred ≡ I(x0; zτ), and the mutual
+information between the current system output and the past ligand concentration trajectory is the past information
+Ipast ≡ I(x0; Lp). The Lagrange multiplier γ sets the relative cost of storing past information over obtaining predictive
+information. Given a value of γ, Eq. C1 is maximized by optimizing the mapping of the past ligand concentration
+trajectory Lp onto the current output x0. Since, by the data processing inequality, we have Ipast ≥ Ipred, for γ = 1
+the objective function is maximized by Ipast = Ipred = 0. As γ is decreased both the past and predictive information
+increase, and the parametric curve in the Ipast − Ipred plane that arises is the information bound. For γ = 0 there
+is no cost to storing past information. The predictive information is then only limited by the amount of information
+contained in the past about the future signal property: Ipred ≤ I(Lp; zτ).
+1.
+Gaussian information bottleneck
+In general equation C1 can be difficult to solve, as all mappings from Lp to X0 are allowed. However, the problem
+becomes analytically tractable when the joint probability distribution of Lp and zτ is a multivariate Gaussian. Here,
+
+14
+we follow the procedure of Chechik and coworkers to obtain this mapping [12]. In the Gaussian model, the optimal
+mapping from Lp to x0 is a linear one [12]
+x0 = ALp + ξ;
+ξ ∼ N(0, σ2
+ξ),
+(C2)
+where A is a row vector which determines how strongly each entry in Lp contributes to the scalar output X0 at any
+point in time. The random variable ξ is the noise added to the signal due to the stochastic nature of the mapping; it is
+a Gaussian random variable independent of Lp with 0 mean and variance σ2
+ξ. Finding the optimal mapping from Lp
+to x0 corresponds to finding the optimal combination of A and σ2
+ξ. It can be shown that for any pair (A, σ2
+ξ), there
+exists a pair (A′, 1) which yields the same values for Ipast and Ipred after maximization of Eq. C1 [12]. Therefore, we
+can set σ2
+ξ = 1 without altering the information curve.
+To obtain the information bound, we rewrite Eq. C1 using the definition of the mutual information between Gaussian
+random variables:
+L = 1
+2 log(σ2
+x/σ2
+x|z) − γ 1
+2 log(σ2
+x/σ2
+x|L),
+(C3)
+with the total variance σ2
+x in the output x0, the output variance conditional on the future signal property σ2
+x|z ≡ σ2
+x|zτ ,
+and the output variance conditional on the complete history of ligand concentrations σ2
+x|L ≡ σ2
+x|Lp. The latter is just
+the variance caused by the intrinsic noise, σ2
+x|L = σ2
+ξ = 1. The total variance in x0 can be expressed in terms of the
+mapping vector A and the variance in the past signal using Eq. C2, σ2
+x = AΣLAT + 1, where ΣL ≡ ΣLp is the
+covariance matrix of the past ligand concentration trajectory Lp. To express the output variance conditional on the
+future signal property zτ we use the Schur complement formula, which in general form reads:
+Σx|y = Σx − ΣxyΣ−1
+y Σyx,
+(C4)
+where Σyx = ΣT
+xy. Using this formula to rewrite σ2
+x|z, and then using the linear relation from Eq. C2 again, we obtain
+σ2
+x|z = AΣL|zAT + 1.
+Filling in the expressions for the variances in L (Eq. C3) gives:
+L = 1
+2
+�
+(1 − γ) log
+���AΣLAT + 1
+���
+− log
+���AΣL|zAT + 1
+����
+.
+(C5)
+For any symmetric matrix C we have
+δ
+δA log |ACAT | =
+�
+ACAT �−1 2AC, such that we obtain for the derivative of
+L to A:
+δL
+δA = (1 − γ)
+AΣL
+AΣLAT + 1 −
+AΣL|z
+AΣL|zAT + 1.
+(C6)
+In our case A is a row vector, and both denominators are thus scalars. We find the maximum of L by equating its
+derivative to 0, which gives:
+AΣL|zΣ−1
+L
+= (1 − γ)AΣL|zAT + 1
+AΣLAT + 1 A.
+(C7)
+For this equality to hold A must either be identically 0, or a left eigenvector of the matrix ΣL|zΣ−1
+L
+with eigenvalue:
+λ = (1 − γ)AΣL|zAT + 1
+AΣLAT + 1 .
+(C8)
+Here, we note that if the signal statistics is sufficiently rich and the prediction complexity sufficiently large (because,
+for example, multiple signal characteristics need to be predicted), then the matrix ΣL|zΣ−1
+L
+has multiple eigenvectors
+with non-trivial eigenvalues 0 < λi < 1 [12]. This reflects the idea that storing the past information that is necessary
+to enable this complex prediction task may require multiple output components, i.e.
+an output vector x, where
+each output component has an integration kernel given by one of the eigenvectors of ΣL|zΣ−1
+L
+[12]. However, for
+Markovian signals only one eigenvector with non-trivial eigenvalue 0 < λ < 1 emerges, which means that one output
+component is sufficient to encode the required information. For the non-Markovian signals studied here, ΣL|zΣ−1
+L
+has two eigenvectors if both the future value and its derivative need to be predicted (and z = (ℓτ, vτ)); to optimally
+predict both features from the current output, two output components are then required, provided Ipast is sufficiently
+
+15
+large. However, here we consider the scenario that only the future derivative needs to be predicted, in which case only
+one non-trivial eigenvector emerges, and one output component is sufficient for encoding the required information.
+We leave the problem of predicting multiple signal features via multiple output components for future work.
+We can define the optimal mapping A = ||A||ν where ν is the normalized left eigenvector of ΣL|zΣ−1
+L corresponding
+to its smallest eigenvalue, 0 < λ < 1. The magnitude can be found by solving Eq. C8 for ||A||, using from Eq. C7
+that λνΣLνT = νΣL|zνT . This gives for the optimal mapping:
+Aopt =
+��
+1−γ−λ
+ν1ΣLνT
+1 λγ ν1
+for
+0 < λ < 1 − γ,
+0
+for
+1 − γ ≤ λ ≤ 1.
+(C9)
+We can substitute ||A||2 = (1 − γ − λ)/(νΣLνT λγ) in the definitions for the mutual information to express them
+in terms of λ and γ. For the past information we obtain:
+Ipast = 1
+2 log
+�
+||A||2νΣLνT + 1
+�
+,
+= 1
+2 log
+�1 − γ
+γ
+1 − λ
+λ
+�
+.
+(C10)
+And for the predictive information:
+Ipred = 1
+2 log
+�
+||A||2νΣLνT + 1
+�
+− 1
+2 log
+�
+||A||2νΣL|ℓτ νT + 1
+�
+,
+= Ipast − 1
+2 log
+�1 − λ
+γ
+�
+,
+= 1
+2 log
+�1 − γ
+λ
+�
+.
+(C11)
+2.
+Markovian signal
+To obtain the information bound for prediction of the future ligand concentration of a Markovian signal, we need
+to determine the eigenvalues and vectors of the matrix (see Eqs. C7 and C8)
+W = ΣL|ℓτ Σ−1
+L .
+(C12)
+Using the Schur complement formula (Eq. C4) to rewrite the conditional matrix gives ΣL|ℓτ = ΣL − ΣLℓτ ΣT
+Lℓτ /σ2
+ℓ.
+Then defining the normalized matrices RL = ΣL/σ2
+ℓ and RLℓτ = ΣLℓτ /σ2
+ℓ we find
+W = IN − RLℓτ RT
+Lℓτ R−1
+L .
+(C13)
+where N is the length of the input trajectory Lp. The correlation matrix of the past trajectory is symmetric with
+entries R(i,j)
+L
+= exp(−|i − j|∆t/τℓ), where ∆t is the discretization timestep of the past trajectory Lp and i ranges
+from 1 to N. This is a Kac-Murdock-Szeg¨o matrix, and its inverse is known:
+R−1
+L
+=
+1
+1 − e2∆t/τℓ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+−e−∆t/τℓ
+0
+. . .
+. . .
+0
+−e−∆t/τℓ 1 + e−2∆t/τℓ
+−e−∆t/τℓ
+. . .
+. . .
+0
+0
+−e−∆t/τℓ
+1 + e−2∆t/τℓ
+...
+. . .
+0
+...
+...
+...
+...
+...
+...
+0
+. . .
+. . .
+−e−∆t/τℓ 1 + e−2∆t/τℓ −e−∆t/τℓ
+0
+. . .
+. . .
+0
+−e−∆t/τℓ
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(C14)
+Note that the inverse matrix is tridiagonal. The length N cross-correlation vector between past trajectory and future
+concentration has entries R(i)
+Lℓτ = exp(−(τ + (i − 1)∆t)/τℓ). The product of the correlation matrices is surprisingly
+simple:
+RLℓτ RT
+Lℓτ R−1
+L
+= e−2τ/τℓ
+�
+�
+�
+�
+�
+1
+0 . . . 0
+e−∆t/τℓ
+0 . . . 0
+...
+...
+...
+...
+e−(N−1)∆t/τℓ 0 . . . 0
+�
+�
+�
+�
+� .
+(C15)
+
+16
+Using this result we can straightforwardly determine the eigenvalues,
+|W − λIN| = 0,
+���������
+�
+�
+�
+�
+�
+1 − λ − e−2τ/τℓ
+0
+. . .
+0
+−e−(τ+∆t)/τℓ
+1 − λ . . .
+0
+...
+...
+...
+...
+−e−(τ+(N−1)∆t)/τℓ
+0
+. . . 1 − λ
+�
+�
+�
+�
+�
+���������
+= 0.
+(C16)
+The only contribution to the determinant comes from the diagonal, and the only nontrivial eigenvalue is thus λ =
+1−e−2τ/τl. The optimal mapping is thus onto a one-dimensional scalar output x0. The corresponding left eigenvector
+is given by
+ν1W = (1 − e−2τ/τl)ν1,
+(C17)
+which holds for ν1 =
+�1 0 . . . 0�
+. The optimal mapping for the prediction of a one-dimensional OU-process is thus
+to copy its most recent value. This agrees with intuition as for any Markovian process, all the information about the
+future signal is contained in the most recent value. For a continuous input signal (rather than a discretized signal),
+and a continuous integration kernel k(t) (rather than a mapping vector A), this means that the optimal integration
+kernel is kopt(t) = aδ(t).
+3.
+Non-Markovian signal
+To find the optimal mapping for the prediction of the derivative of a non-Markovian signal, based on its history of
+ligand concentrations, we need to find the eigenvalues and vectors of the matrix
+W = ΣL|vτ Σ−1
+L ,
+= IN − 1
+σ2v
+ΣLvτ ΣT
+Lvτ Σ−1
+L .
+(C18)
+The covariance matrix of the past trajectory is symmetric with entries Σ(i,j)
+L
+= ⟨δℓ(0)δℓ(|i − j|∆t)⟩ where both i and
+j range from 1 to N, the past trajectory length. The covariance vector between past trajectory and future derivative
+has entries Σ(i,j)
+Lvτ = ⟨δℓ(0)δv(τ + (i − 1)∆t)⟩. Both the concentration auto-correlation function, and the concentration
+to future derivative cross-correlation function, are shown in Eq. B7.
+To better understand the optimal mapping of this signal we numerically investigate the eigenvalues of the matrix
+W . For the prediction of vτ, there is only one non-trivial eigenvalue. Like for the Markovian signal, this shows that
+for the prediction of the derivative of this non-Markovian signal, the optimal mapping is always onto a scalar output.
+The non-trivial eigenvalue λ decreases with the discretization timestep ∆t and is minimal for ∆t → 0 Fig. 5. In this
+limit, λ has the same magnitude for any N ≥ 2, see Fig. 5. A smaller eigenvalue λ corresponds to larger past and
+predictive information and a larger ratio Ipred/Ipast (Eq. C10 and Eq. C11), given any value of the Lagrange multiplier
+γ. For the optimal mapping we must thus have N ≥ 2 and ∆t → 0, where N sets both the past trajectory and the
+mapping vector length. Because increasing the length above two does not yield an improvement in the value of λ1 we
+focus on N = 2.
+The fact that to reach the optimum we must have N = 2 and ∆t → 0, shows that the optimal kernel A takes an
+instantaneous measurement of a combination of the most recent ligand concentration, and its derivative. This can be
+understood as follows, for a trajectory of length two, the mapping vector also has length two, A = ||A||( ˆw1, ˆw2), with
+�
+ˆw2
+1 + ˆw2
+2 = 1. We can then express the linear mapping of Lp to x0 (Eq. C2) as:
+x0 = ||A||
+�
+( ˆw1 + ˆw2)δℓ(0) − ˆw2∆tδℓ(0) − δℓ(−∆t)
+∆t
+�
++ ξ,
+(C19)
+This expression shows that, as ∆t → 0, the two entries of A combine both the most recent signal value and the most
+recent derivative to generate x0. This is intuitive because the signal is completely defined by its concentration and
+derivative (Eq. B3). For this reason, and to obtain analytical insight into the optimal weights, we inspect the final
+two entries of the past ligand concentration trajectory in the limit ∆t → 0, which defines the past signal in terms of
+its most recent concentration and derivative
+S0 ≡
+�δℓ(0) v(0)�T .
+(C20)
+
+17
+0.05
+0.10
+0.15
+0.20
+0.560
+0.565
+0.570
+0.575
+0.580
+Discretization timestep Δt
+Smallest eigenvalue λ
+N = 2
+N = 3
+N = 4
+FIG. 5. The smallest eigenvalue of the IB matrix is minimal for N ≥ 2 and ∆t → 0. A smaller eigenvalue corresponds
+to a larger ratio Ipred/Ipast for any given value of the Lagrange multiplier γ. Parameters: friction timescale τ −1
+v
+= 0.862s−1 as
+determined in [18], prediction interval τ = τv, and ω0 = 0.4s−1 such that the system is slightly overdamped.
+Because the signal is Markovian in the joint properties δℓ and v, the vector S0 contains the same information as the
+trajectory Lp. The past information is now the mutual information between x0 and S0, i.e. Ipast = I(x0; S0). The
+output x0 can then also be written as a projection of S0 via the alternative mapping vector ˜
+A = ||A||(ˆa,ˆb):
+x0 = ||A||
+�
+ˆaδℓ(0) + ˆbv(0)
+�
++ ξ.
+(C21)
+Comparison with Eq. C19 shows how the components of ˜
+A relate back to those in A,
+ˆw1 = ˆa + ˆb/∆t,
+(C22)
+ˆw2 = −ˆb/∆t.
+(C23)
+To obtain the optimal mapping vector ˜
+A the matrix of signal statistics of which the eigenvalues and -vectors need
+to be determined is
+W = Σs|vτ Σ−1
+s ,
+(C24)
+with
+Σs =
+�
+σ2
+ℓ
+0
+0
+σ2
+v
+�
+,
+(C25)
+Σs|vτ = Σs − 1
+σ2v
+Σsvτ ΣT
+svτ ,
+(C26)
+Σsvτ =
+�
+⟨δℓ(0)δv(τ)⟩
+⟨δv(0)δv(τ)⟩
+�
+.
+(C27)
+We thus obtain
+W = I −
+�
+�
+�
+⟨δℓ(0)δv(τ)⟩2
+σ2
+ℓ σ2v
+⟨δℓ(0)δv(τ)⟩⟨δv(0)δv(τ)⟩
+σ4v
+⟨δℓ(0)δv(τ)⟩⟨δv(0)δv(τ)⟩
+σ2
+ℓ σ2v
+⟨δv(0)δv(τ)⟩2
+σ4v
+�
+�
+� .
+(C28)
+This matrix has one nontrivial eigenvalue, λ = 1 − ⟨δv(0)δv(τ)⟩2
+σ4v
+− ⟨δℓ(0)δv(τ)⟩2
+σ2
+ℓ σ2v
+, which depends on the normalized
+correlation functions between on the one hand the current concentration or derivative, and on the other hand the
+future derivative. The corresponding left eigenvector is
+ν1 = Q−1 �
+1
+σℓ
+⟨δℓ(0)δv(τ)⟩
+σℓσv
+1
+σv
+⟨δv(0)δv(τ)⟩
+σ2
+v
+�
+,
+(C29)
+where Q normalizes the vector.Using the linear mapping x0 = ||A||ν1S0 + ξ, and defining G ≡ ||A||/Q, shows that
+the optimal output should be generated as follows
+xopt
+0
+= G
+�⟨δℓ(0)δv(τ)⟩
+σℓσv
+δℓ(0)
+σℓ
++ ⟨δv(0)δv(τ)⟩
+σ2v
+v(0)
+σv
+�
++ ξ.
+(C30)
+
+18
+Clearly, the optimal mapping depends on the (normalized) cross-correlation coefficient ρℓ0vτ ≡ ⟨δℓ(0)δv(τ)⟩/(σℓσv)
+between the current concentration δℓ(0) and future derivative δv(τ), and the cross-correlation coefficient ρv0vτ between
+the current derivative δv(0) and future derivative δv(τ). Indeed, to optimally predict the future derivative, the cell
+should also use the current concentration and not only its current derivative. However, in the limit that the range of
+concentrations sensed becomes very large, corresponding to ω0 → 0, the current concentration is no longer correlated
+with the future derivative, and ρℓ0vτ → 0 (Eq. B8). In this limit, ˆa = 0 and ˆb = 1, and the kernel becomes a perfectly
+adaptive, derivative-taking kernel:
+lim
+ω0→0 xopt
+0
+= ||A||v(0) + ξ.
+(C31)
+If we translate this back to the vector ||A||( ˆw1, ˆw2), operating on a ligand concentration trajectory Lp, the optimal
+weights become ˆw1 = − ˆw2.
+Appendix D: Past and predictive information for linear signalling networks
+In order to address how close biochemical networks can come to the information bounds derived above, we here
+describe how we obtain the past and predictive information for any linear (biochemical) network.
+We then use
+the resulting general expressions to compute the past and predictive information for the push-pull network and the
+chemotaxis system of the main text.
+For any linear network the output can be written as
+δx(t) =
+� t
+−∞
+ds k(t − s)δℓ(s) + ηx(t).
+(D1)
+The mapping kernel k(t) is a property of the network and describes how the input signal is mapped onto the output.
+The noise term ηx(t) is a sum of convolutions over all white noise processes in the network and corresponding network
+mapping functions, see Eq. A2. The variance in the output can generally be split up in a part caused by the signal
+and a part cause by the noise, and we have
+σ2
+x =
+� t
+−∞
+ds
+� t
+−∞
+ds′k(t − s)k(t − s′)⟨δℓ(s)δℓ(s′)⟩ + σ2
+ηx,
+= σ2
+x|η + σ2
+x|L,
+(D2)
+where σ2
+x|η is the signal variance, i.e. all noise terms are fixed, and σ2
+x|L is the noise variance, i.e. the complete history
+of the signal is fixed. Using this decomposition we find for the past information, which is the mutual information
+between the current output and the complete signal history,
+Ipast(x0; Lp) = 1
+2 log
+�
+σ2
+x
+σ2
+x|L
+�
+= 1
+2 log(1 + SNR),
+(D3)
+where the signal-to-noise ratio is defined as SNR = σ2
+x|η/σ2
+x|L. Using the same definition for the mutual information
+when deriving the predictive information between current output and future ligand concentration, we obtain
+Ipred(x0; ℓτ) = 1
+2 log
+�
+σ2
+x
+σ2
+x|ℓτ
+�
+,
+= 1
+2 log
+�
+1 +
+σ2
+x|η
+σ2
+x|L
+�
+− 1
+2 log
+�
+1 +
+σ2
+x|η − ⟨δx(0)δℓ(τ)⟩2/σ2
+ℓ
+σ2
+x|L
+�
+,
+= Ipast − 1
+2 log(1 + cSNR).
+(D4)
+In the second line we used the Schur complement formula, Eq. C4, to decompose the variance in the output conditioned
+on the future signal: σ2
+x|ℓτ = σ2
+x−⟨δx(0)δℓ(τ)⟩2/σ2
+ℓ. The quantity σ2
+x|η−⟨δx(0)δℓ(τ)⟩2/σ2
+ℓ can be understood as follows:
+the first term σ2
+x|η is the contribution to the total variance of the output σ2
+x that comes from the signal variations,
+while the second term quantifies the variance in the output that is correlated with the future input. The difference is
+thus the variance in the output coming from the signal variations that are not correlated with the future input. The
+
+19
+ratio in the second logarithm can thus be understood as a conditional SNR that quantifies the part of the signal to
+noise ratio that does not contain information about the future signal. This becomes more clear when considering its
+form in terms of the mapping kernel and signal correlation functions. For any linear signalling network we have
+σ2
+x|η − ⟨δx(0)δℓ(τ)⟩2/σ2
+ℓ =
+� 0
+−∞
+ds
+� 0
+−∞
+ds′k(−s)k(−s′)
+�
+⟨δℓ(s)δℓ(s′)⟩ − ⟨δℓ(τ)δℓ(s′)⟩⟨δℓ(τ)δℓ(s)⟩
+σ2
+ℓ
+�
+,
+(D5)
+where the term in parentheses is the conditional variance in the past signal trajectory given a future value, ΣL|ℓτ .
+The form in Eq. D4 thus tells us that the predictive information is equal to the past information, minus the bits
+that do not contain information about the future ligand concentration. This difference is indeed the part of the past
+information that does contain predictive information about the future signal.
+Although the expression above (Eq. D4) nicely relates the past and predictive information, a more straightforward
+way of obtaining the predictive information is by expressing it directly in terms of the correlation between the current
+output and the future ligand concentration:
+Ipred(x0; ℓτ) = 1
+2 log
+�
+σ2
+x
+σ2
+x|ℓτ
+�
+= −1
+2 log
+�
+1 − ⟨δx(0)δℓ(τ)⟩2
+σ2xσ2
+ℓ
+�
+,
+(D6)
+where we again used the Schur complement formula to rewrite σ2
+x|ℓτ . Written this way we thus see that the predictive
+information depends on the normalized correlation between the current network output and the future ligand con-
+centration. We can simply exchange the future ligand concentration for the future derivative when considering the
+chemotaxis network.
+To compute the past information for linear signalling networks we use Eq. D3, and we thus need to compute the
+SNR. To compute the predictive information for the prediction of a future ligand concentration, we need to compute
+the ‘future correlation function’ ⟨δx(0)δℓ(τ)⟩. For the prediction of the future derivative we need ⟨δx(0)δv(τ)⟩.
+Appendix E: Push-pull network
+We consider a push-pull network that consists of a phosphorylation-dephosphorylation cycle downstream of a
+receptor.
+When bound to ligand, the receptor itself or its associated kinase, such as CheA in E. coli, catalyzes
+the phosphorylation of a readout protein x, like CheY. Active readout molecules x∗ can decay spontaneously or be
+deactivated by an enzyme (phosphatase), such as CheZ in E. coli. This cycle is driven by the turnover of fuel such as
+ATP. We recognize that inside the living cell, the chemical driving is typically large: for example, the free energy of
+ATP hydrolysis is about 20kBT, which means that the system essentially operates in the irreversible regime [10, 36].
+This system consists of the following reactions:
+R + L
+k+
+−−⇀
+↽−−
+k−
+RL
+(E1)
+RL + x
+kf
+−−→ RL + x∗
+(E2)
+X∗
+kr
+−−→ X
+(E3)
+Both the total number of receptors RT = R + RL and read-out molecules XT = X + X∗ are conserved moieties. The
+chemical Langevin equations of this system are:
+˙
+RL = [RT − RL(t)]ℓ(t)k+ − RL(t)k− + Bc(RL, ℓ)ξc(t),
+(E4)
+˙x∗ = [XT − x∗(t)]RL(t)kf − x∗(t)kr + Bx(RL, x∗)ξx(t),
+(E5)
+where RL is the number of bound receptors, x∗ the number of phosphorylated read-out molecules, and ξi denote
+independent Gaussian white noise with unit variance, ⟨ξi(t)ξj(t′)⟩ = δijδ(t − t′). The noise strengths are Bc(RL, ℓ) =
+�
+(RT − RL(t))ℓ(t)k+ + RL(t)k− and Bx(RL, x∗) =
+�
+(XT − x∗(t))RL(t)kf + x∗(t)kr. The steady-state fraction of
+ligand-bound receptors is p ≡ RL/RT = ¯ℓ/(¯ℓ+KD) with the dissociation constant KD = k−/k+, and the steady-state
+fraction of phosphorylated readout molecules is f ≡ ¯x∗/XT = pRT/(pRT + kr/kf).
+In the linear-noise approximation, expanding Eqs. E4 and E5 to first order around their steady state, the equations
+become
+δ ˙
+RL = b δℓ(t) − δRL(t)/τc + ηc(t),
+(E6)
+δ ˙x∗ = γ δRL(t) − δx∗(t)/τr + ηx(t).
+(E7)
+
+20
+The parameters b = RTp(1 − p)/(¯ℓτc) and γ = XTf(1 − f)/(RTpτr) are effective rates of receptor-ligand binding
+and readout phosphorylation, respectively.
+The decay rate of correlations in the receptor-ligand binding state is
+τc−1 = ¯ℓk+ + k−, and that of the readout phosphorylation state is τr−1 = pRTkf + kr. The rescaled white noise
+processes have strengths ⟨η2
+c⟩ = B2
+c = 2RTp(1 − p)/τc and ⟨η2
+x⟩ = B2
+x = 2XTf(1 − f)/τr.
+1.
+Model statistics
+The relevant quantity to compute the past information is the variance in the output, decomposed into the part
+caused by signal variation and the part caused by noise. To compute the predictive information we further need the
+correlation function between the current output and a future ligand concentration ⟨δℓ(τ)δx∗(0)⟩. These quantities
+can be obtained via their Fourier transforms, as in Eq. A5 and Eq. A6. The matrices describing the properties of the
+signalling network are, as defined below Eq. A1,
+G =
+�
+b
+0
+�
+,
+(E8)
+J =
+�
+−τc−1
+0
+γ
+−τr−1
+�
+,
+(E9)
+B =
+��
+⟨η2c⟩
+0
+0
+�
+⟨η2x⟩
+�
+=
+��
+2RTp(1 − p)/τc
+0
+0
+�
+2XTf(1 − f)/τr
+�
+.
+(E10)
+A useful property of the network is the matrix exponential of its Jacobian, which in Fourier space is (see Eq. A2
+and Eq. A4)
+F{eJ t}(ω) = (iωI2 − J )−1,
+=
+�
+�
+1
+1/τc+iω
+0
+γ
+(1/τc+iω)(1/τr+iω)
+1
+1/τr+iω
+�
+� .
+(E11)
+We then have G(ω) = F{eJ t}(ω)G and N(ω) = F{eJ t}(ω)B, see also Eq. A4 and Eq. A5. The integration kernel that
+maps the ligand concentration onto the output of the push-pull network, see Eq. D1, is given by the inverse Fourier
+transform of the second entry of G(ω), which is the frequency dependent gain, ˜gℓ→x(ω), from ℓ to x:
+k(t) ≡ F−1{˜gℓ→x(ω)} = bγτcτr
+1
+τr − τc
+�
+e−t/τr − e−t/τc�
+,
+= XTf(1 − f)(1 − p)/¯ℓ
+1
+τr − τc
+�
+e−t/τr − e−t/τc�
+,
+(E12)
+The so-called static gain of the network is the integral of this kernel over all time, ¯gℓ→x ≡
+� ∞
+0
+k(t)dt = XTf(1 −
+f)(1 − p)/¯ℓ.
+This parameter quantifies how much a step change in the input concentration changes the steady-
+state level of the output: ¯gℓ→x = ∂ ¯x∗/∂¯ℓ. We will use this parameter in the statistical quantities that follow. The
+static gain is also given by ¯gℓ→x = ¯gℓ→RL¯gRL→x, with ¯gℓ→RL = p(1 − p)RT/¯ℓ the static gain from ¯ℓ to RL and
+¯gRL→x = f(1 − f)XT/(pRT) the static gain from RL to x∗.
+We model the Markovian ligand concentration as a 1-dimensional OU process Eq. B1, which has the following
+power spectrum
+Sℓ(ω) = ⟨|δℓ(ω)|2⟩ =
+2σ2
+ℓ/τℓ
+1/τ 2
+ℓ + ω2 .
+(E13)
+This yields the following expression for the power spectra (see Eq. A5):
+G(−ω)Sℓ(ω)G(ω)T = b2
+�
+�
+1
+1/τc2+ω2
+γ
+1
+1/τr−iω
+1
+1/τc2+ω2
+γ
+1
+1/τr+iω
+1
+1/τc2+ω2 γ2
+1
+1/τr2+ω2
+1
+1/τc2+ω2
+�
+�
+2σ2
+ℓ/τℓ
+1/τ 2
+ℓ + ω2
+(E14)
+|N(ω)|2 = ⟨η2
+c⟩
+�
+�
+�
+1
+1/τc2+ω2
+γ
+1
+1/τr−iω
+1
+1/τc2+ω2
+γ
+1
+1/τr+iω
+1
+1/τc2+ω2 γ2
+1
+1/τr2+ω2
+1
+1/τc2+ω2 + ⟨η2
+x⟩
+⟨η2c⟩
+1
+1/τr2+ω2
+�
+�
+�
+(E15)
+
+21
+We thus have for the power spectrum of the read-out:
+Sx(ω) = ˜g2
+ℓ→x(ω)Sℓ(ω) + N 2
+x(ω)
+=
+2b2γ2σ2
+ℓ/τℓ
+(1/τr2 + ω2)(1/τc2 + ω2)(1/τ 2
+ℓ + ω2) +
+γ2⟨η2
+c⟩
+(1/τr2 + ω2)(1/τc2 + ω2) +
+⟨η2
+x⟩
+1/τr2 + ω2 ,
+(E16)
+The variance in the read-out σ2
+x = 1/(2π)
+� ∞
+−∞ Sx(ω) is hence given by
+σ2
+x = σ2
+x|η + σ2
+x|L
+= ¯g2
+ℓ→x
+1 + τr/τℓ + τr/τc
+(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc)σ2
+ℓ + ¯g2
+RL→xRTp(1 − p)
+1
+1 + τr/τc
++ XTf(1 − f),
+= ¯g2
+ℓ→x
+1 + τr/τℓ + τr/τc
+(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc)
+�
+��
+�
+dynamical gain
+σ2
+ℓ + XTf(1 − f)
+�
+1 + ¯gℓ→x
+¯ℓ
+RTp
+1
+1 + τr/τc
+�
+,
+(E17)
+where ¯gRL→x = γτr = XTf(1 − f)/(RTp) is the static gain from the receptor to the readout. The expression above
+gives insight into the role of the different network components in shaping the noise in the readout. It can be seen
+that the contribution from the signal variance σ2
+ℓ to σ2
+x is determined by the static gain ¯g2
+ℓ→x, which is proportional
+to XT, and a factor that only depends on ratios of timescales. Their product is the dynamical gain, which decreases
+monotonically with τr.
+The intrinsic noise in the phosphorylation state of the read-outs leads to the noise term
+XTf(1 − f), which cannot be averaged out. The noise arising from ligand binding and unbinding increases with the
+static gain, but can be mitigated by increasing the number of receptors or the integration time τr. The latter strategy
+is what we call time-averaging.
+The signal-to-noise ratio SNR = σ2
+x|η/σ2
+x|L can straightforwardly be obtained from Eq. E17. This is the quantity
+that sets the magnitude of the past information, see Eq. D3.
+To determine the predictive information we need
+to compute the correlation function from the current output to the future ligand concentration ⟨δx(0)δℓ(τ)⟩. This
+requires the cross-spectrum from output to ligand concentration, which is given by (Eq. A6)
+˜gℓ→x(−ω)Sℓ(ω) =
+bγ
+(1/τc − iω)(1/τr − iω)
+2σ2
+ℓ/τℓ
+1/τ 2
+ℓ + ω2 .
+(E18)
+From this power spectrum we obtain the required correlation function by taking the inverse Fourier transfrom:
+⟨δx(0)δℓ(τ)⟩ = F−1{˜gℓ→x(−ω)Sℓ(ω)},
+=
+¯gℓ→xσ2
+ℓ
+(1 + τc/τℓ)(1 + τr/τℓ)e−τ/τℓ.
+(E19)
+This correlation function thus decays exponentially with the prediction interval τ at a rate τ −1
+ℓ
+, just as the signal auto-
+correlation. The (squared) correlation coefficient, which sets Ipred, is given by ⟨δx(0)δℓ(τ)⟩2/(σ2
+ℓσ2
+x) = ρ2
+ℓxe−2τ/τℓ,
+with the (squared) instantaneous correlation coefficient (for convenience given as its inverse)
+ρ−2
+ℓx =
+¯ℓ2
+σ2
+ℓ
+�
+1 + τr
+τℓ
+�2 �
+1 + τc
+τℓ
+�2 �
+1
+XT f(1 − f)(1 − p)2 +
+1
+RT p(1 − p)(1 + τr/τc) + σ2
+ℓ
+¯ℓ2
+1 + τr/τℓ + τr/τc
+(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc)
+�
+.
+(E20)
+When the right-hand-side is minimized, the correlation is thus maximized. This expression shows that increasing XT
+and RT always increases the instantaneous correlation coefficient, and that the fraction of phosphorylated readout
+molecules in steady state that maximizes the correlation coefficient is f = 1/2.
+2.
+Past and predictive information of the push-pull network
+Using the quantitites computed above, we can determine both the past and the predictive information. For the
+past information we use Eq. D3, whith the SNR from Eq. E17:
+SNR = σ2
+x|η/σ2
+x|L = (1 − p)σ2
+ℓ
+¯ℓ2
+1 + τr/τℓ + τr/τc
+(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc)
+� �
+1
+XTf(1 − f)(1 − p) +
+1
+RTp(1 + τr/τc)
+�
+.
+
+22
+The predictive information is a function of the correlation between the current output and the future ligand concen-
+tration, Eq. D6. This correlation can be decomposed into the instantaneous correlation coefficient and an exponential
+decay on the timescale of the ligand concentration fluctuations, Eq. E19. We thus obtain for the predictive information,
+Ipred(x0; ℓτ) = −1
+2 log(1 − ρ2
+ℓxe−2τ/τℓ).
+(E21)
+The instantaneous correlation coefficient ρ2
+ℓx is given in Eq. E20. From Eq. E21 it also becomes clear that while
+the value of the predictive information depends on the forecast interval τ, the optimal design of the network that
+maximizes the predictive information, determined by the optimal ratio XT/RT, the optimal integration time τr, and
+the optimal ligand-bound receptor fraction p, does not depend on the forecast interval τ.
+3.
+Optimal resource allocation
+Increasing the number of receptor or readout molecules always increases the precision with which the cell can predict
+a signal (see Eq. E20). However, when the total resource pool is constrained, the cell has to choose whether it makes
+more receptors or more readout molecules. To find the optimal ratio of read-out to receptor molecules we, can use
+the C = ART + BXT to express XT and RT in terms of the total cost C and the ratio XT/RT:
+XT = C
+XT/RT
+A + BXT/RT
+,
+(E22)
+RT = C
+1
+A + BXT/RT
+.
+(E23)
+The factors A and B set the cost of receptors and readout molecules, respectively. Substituting these expressions for
+XT and RT into the expression for the correlation coefficient between the output and ligand concentration (Eq. E20),
+setting the derivative of the resulting expression with respect to XT/RT to zero, and solving for XT/RT gives
+(XT/RT)opt =
+��
+1 + τr
+τc
+�
+p
+1 − p
+1
+f(1 − f)
+A
+B ,
+= 2
+�
+p/(1 − p)
+�
+1 + τr/τc,
+(E24)
+where for the second line we used A = B = 1 and f = f opt = 1/2. This is the optimal ratio of readout to receptor
+molecules in the push-pull network, given an integration time τr and a steady state fraction of ligand-bound receptors
+p.
+Perhaps surprisingly, this optimal ratio (XT/RT)opt maximizes, for a given τr and p, not only the predictive
+information, but also the past information. This is because the ratio XT/RT determines, together with τr and p, the
+interval ∆ for sampling the ligand-binding state of the receptor: when the ratio XT/RT obeys Eq. E24, the readout
+molecules sample each receptor molecule roughly once every correlation time: ∆ ∼ τc [10, 36]. Eq. E24 is thus a
+statement about optimally extracting the information that is encoded in the receptor-ligand binding history, both
+concerning the past information and the predictive information. This is illustrated in Fig. 6.
+4.
+Operating costs diverge when approaching the information bound
+The precision of any sensing device is limited by the resources that are devoted to it. The cost function we consider
+in this work is
+C = λ(RT + XT) + c1XT∆µ/τr.
+(E25)
+The first term is the maintenance cost; this is the cost of producing new network components at the growth rate
+λ. The second term is the operating cost and describes the chemical power that is necessary to run the network; it
+depends on the flux through the network, XT/τr, and the free-energy drop ∆µ over a full cycle of phosphorylation
+and dephosphorylation, given by the free energy of ATP hydrolysis. The coefficient c1 describes the relative energetic
+cost of synthesising the components during the cell cycle, versus that of running the system. In the main text we
+consider the case where c1 → 0. Here we will investigate how close cells can come to the information bound when c1
+is finite, thus including the chemical power cost of running the network.
+It is clear from Eq. E25 that for finite c1 the operating cost diverges when τr → 0. Because the optimal IBM
+solutions are instantaneous, this is precisely the limit in which the network must be to reach the information bound.
+
+23
+0.0
+0.5
+1.0
+1.5
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Past info (bits)
+Predictive info (bits)
+τr=0.01
+τr=0.2
+τr=0.5
+τr=1
+Varied XT/RT
+p=0.1
+p=0.2
+p=0.4
+FIG. 6.
+The past and predictive information are maximized by the same ratio XT/RT and fraction p.
+The
+information plane, showing the information bound in black, and the isocost line C = 104 in gray. To construct the coloured
+lines in this figure the ratio XT/RT has been varied from zero to a value beyond the optimal value that maximizes Ipast and
+Ipred. This is done for several values of the receptor occupancy p (p = 0.1 in red, p = 0.2 in blue, p = 0.4 in orange), and
+for several values of τr (indicated in the figure). When XT/RT reaches its optimal value, both Ipast and Ipred are maximal.
+When the ratio is increased further the system moves back to the origin via the same coordinates. Only the integration time
+τr meaningfully distinuishes between strategies that maximize predictive or past information, or that approach the information
+bound. The reason is that XT/RT, together with τr and p, control the optimal extraction of information that is encoded in
+the receptor-ligand binding history, both concerning Ipast and Ipred. The gray isocost line is obtained by varying τr, while
+maximizing for each τr the correlation coefficient given by Eq. E20; the latter is done by substituting Eq. E24 into Eq. E20 and
+numerically optimizing the resulting expression over p. The isocost line gives the region of Ipast and Ipred that is accessible for
+a given resource cost C. Parameter values are A = B = 1, f = 1/2, (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.
+As a consequence, when we consider the operating costs, the push-pull network can only be at the information bound
+when (Ipast, Ipred) → (0, 0) or C → ∞ (Fig. 7A). The system can mitigate the operating costs by decreasing XT,
+because this decreases the flux through the cycle. However, this also decreases the gain and thus, eventually, any
+information transduced through the network. In the limit that both XT and τr approach zero, the system approaches
+the information bound at the origin, see both Fig. 7A and B. More generally, when the running costs are taken into
+account, the system time averages more (i.e., τr rises), because frequent measurements are now even more costly. Still,
+τr decreases as the total resource availability C grows.
+Appendix F: Chemotaxis network
+The evidence is mounting that in the E. coli chemotaxis system, receptors cooperatively control the activity of the
+kinase CheA [29, 44–46]. Furthermore, the kinase activity is adaptive due to the methylation of inactive receptors
+[15, 47]. A widely used approach to describe the effects of receptor cooperativity and methylation on kinase activity,
+has been to employ the Monod-Wyman-Changeux (MWC) model [18, 24, 29, 33, 42, 43, 48, 49]. We will follow this
+approach and, more specifically, model the chemotaxis system as described by Tu and colleagues [30]. In this model,
+each receptor can switch between an active and inactive conformational state. Moreover, receptors are partitioned
+into clusters of equal size N. In the spirit of the MWC model, receptors within a cluster switch conformation in
+concert, so that each cluster is either active or inactive [33]. Furthermore, it is assumed that receptor-ligand binding
+and conformational switching are faster than the other timescales in the system. The probability for the kinase, i.e.
+the receptor cluster, to be active, is then described by:
+a(ℓ, m) =
+1
+1 + exp(∆FT (ℓ, m)),
+(F1)
+where ∆FT (ℓ, m) is the total free-energy difference between the active and inactive state, which is a function of the
+ligand concentration ℓ(t) and the methylation level of the cluster m(t). The simplest model adopted here assumes
+a linear dependence of the total free-energy difference on the free-energy difference arising from ligand binding and
+methylation:
+∆FT (ℓ, m) = −∆E0 + N(∆Fℓ(ℓ) + ∆Fm(m)),
+(F2)
+
+24
+A
+B
+FIG. 7. Due to diverging operating costs the push-pull network only reaches the information bound for infinite
+resource availability. (A) In green, the region of accessible predictive and past information in the push-pull network under
+a resource constraint C = λ(RT + XT) + c1XT∆µ/τr, with λ = 1 and c1 = 1/∆µ, corresponding to a cell doubling time of
+roughly 20min [10]. The black line is the information bound; the red and blue dots mark the points where Ipred and Ipast are
+maximized, respectively, under a resource constraint C; the red and blue lines connect these points, respectively, for increasing
+C. The accesible region for C ≤ 104 and the isocost lines for C = 103 and C = 105 have been obtained as described under
+Fig. 6. The forecast interval has been set to one signal correlation time in the future: τ = τℓ. (B) The integration time over
+the receptor correlation time, τr/τc, and the ratio of the number of readout and receptor molecules, XT/RT, as a function of
+the distance θ along the iscocost line for C = 104 in panel A. For θ → 0, both τr and XT go to zero, thus reducing both Ipast
+and Ipred to zero. Other parameter values in both panels are f = f opt = 1/2, (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.
+where the free-energy difference due to ligand binding is
+∆Fℓ(ℓ) = ln(1 + ℓ(t)/KI
+D) − ln(1 + ℓ(t)/KA
+D).
+(F3)
+Between the two states the cluster has an altered dissociation constant, which is denoted KI
+D for the inactive state,
+and KA
+D for the active state. The free-energy difference due to methylation has been experimentally shown to depend
+approximately linearly on the methylation level [29]:
+∆Fm(m) = ˜α( ¯m − m(t)).
+(F4)
+We assume that inactive receptors are irreversibly methylated, and active receptors irreversibly demethylated, with
+zero-order ultrasensitive kinetics [30, 31, 50]. The dynamics of the methylation level of the ith receptor cluster is then
+given by:
+˙mi =(1 − ai(ℓ, mi))kR − ai(ℓ, mi)kB + Bmi(ai)ξ(t),
+(F5)
+with B(i)
+m (ai) =
+�
+(1 − ai(ℓ, mi))kR + ai(ℓ, mi)kB, and unit white noise ξ(t). These dynamics indeed give rise to
+perfect adaptation, since from this equation we find that the steady state cluster activity is given by p ≡ ¯a =
+1/(1 + kB/kR), thus indeed independent of the ligand concentration.
+Finally, active receptors catalyze phosphorylation of read-out molecules, and phosphorylated read-out molecules
+decay at a constant rate. We have
+˙x∗ =
+RT
+�
+i=1
+ai(t)(XT − x∗(t))kf − x∗(t)kr + Bx(ai, x∗)ξ(t),
+(F6)
+where RT is the total number of receptor clusters. The steady state fraction of phosphorylated read-outs is given by
+f ≡ ¯x∗/XT = (1 + kr/(kfRTp))−1.
+
+25
+1.
+Linear dynamics
+We again do a first order approximation around the steady state, defining all variables in terms of deviations from
+their mean: δℓ(t) = ℓ(t) − ¯ℓ, δm(t) = m(t) − ¯m and δa(t) = a(t) − p. The linear form of this model has previously
+been studied in for example [30] and [31]. We obtain for the linear dynamics of the ith cluster activity
+δai(t) = αδmi(t) − βδℓ(t),
+(F7)
+with α = ˜αNp(1 − p) and β = κNp(1 − p), with κ = (¯ℓ + KI
+D)−1 − (¯ℓ + KA
+D)−1. For the methylation on the ith cluster
+and for the readout dynamics we then obtain, as a function of δa(t),
+˙
+δmi = −δai(t)/(ατm) + ηmi(t),
+(F8)
+˙
+δx∗ = γ
+RT
+�
+i=1
+δai(t) − δx∗(t)/τr + ηx(t),
+(F9)
+where we have introduced the relaxation times τm = (α(kR + kB))−1 for methylation and τr = (RTpkf + kr)−1 for
+phosphorylation.
+We have further defined the rate at which an active cluster phosphorylates the readout CheY:
+γ = XTf(1 − f)/(pRTτr). Substituting the expression for δai in Eq. F7 into Eqs. F8 and F9, and expressing the
+dynamics in terms of the methylation on all clusters gives
+d
+dt
+� RT
+�
+i=1
+δmi
+�
+= −
+RT
+�
+i=1
+δmi/τm + qδℓ(t)/(ατm) + ηm(t),
+(F10)
+˙
+δx∗ = −δx∗(t)/τr − γqδℓ(t) + γα
+RT
+�
+i=1
+δmi(t) + ηx(t),
+(F11)
+with q = RTβ (see Eq. F7 for β). The rescaled white noise ηm is the sum of the methylation noise on all receptor
+clusters, ⟨η2
+m⟩ = 2RTp(1 − p)/(ατm), where we have assumed that the methylation noise on the respective receptor
+clusters is independent. The phosphorylation noise has strength ⟨η2
+x⟩ = 2XTf(1 − f)/τr.
+2.
+Parameter values
+A large body of work has studied the parameters of the MWC model for the E. coli chemotaxis system. We have
+listed the parameters relevant for our model in table I. We choose the background concentration ¯ℓ to be in between
+KI
+D and KA
+D, at ¯ℓ = 100µM.
+In this work we analyze the impact of the methylation timescale τm, and the numbers of receptor clusters and
+readout molecules RT and XT, on the past and predictive information. We therefore do not set them to a fixed value,
+but experimental estimates are listed in table II.
+3.
+Model statistics
+Again we take the power spectrum route to determine the variance in the network output, the SNR, and the
+correlation coefficient between current output and the future signal.
+We consider the system to sense the non-
+Markovian ligand concentration defined in equation Eq. B3. Such a signal is characterized by both its concentration
+TABLE I. Measured E. coli chemotaxis parameter values.
+Parameter
+Value
+Source
+Description
+KI
+D
+18µM
+[42, 43]
+MeAsp-Tar dissociation constant inactive receptor
+KA
+D
+2900µM
+[42, 43]
+MeAsp-Tar dissociation constant active receptor
+N
+∼ 6
+[29, 42, 43, 51]
+Number of receptors per cluster
+˜α
+2kBT
+[29]
+Free energy change per added methyl group
+p
+1
+3, 1
+2
+[29, 43]
+Steady state activity at 22◦C, 32◦C
+τr
+∼ 0.1s
+[10, 18, 51]
+Phosphorylation timescale
+
+26
+TABLE II. Approximate E. coli chemotaxis timescales and abundances.
+Parameter
+Value
+Source
+Description
+τm
+∼ 10s
+[15, 18, 29]
+Adaptation time
+Tsr+Tar
+14000, 3300
+[35]
+Rich medium; RP437, OW1 strain
+Tsr+Tar
+24000, 37000
+[35]
+Minimal medium; RP437, OW1 strain
+CheY
+8200, 1400
+[35]
+Rich medium; RP437, OW1 strain
+CheY
+6300, 14000
+[35]
+Minimal medium; RP437, OW1 strain
+and derivative, and the (cross-)power spectra of these properties are
+Ss(ω) =
+�
+Sℓ(ω)
+Sℓ→v(ω)
+Sv→ℓ(ω)
+Sv(ω)
+�
+=
+�
+Sℓ(ω)
+iωSℓ(ω)
+−iωSℓ(ω) ω2Sℓ(ω)
+�
+,
+(F12)
+with
+Sℓ(ω) =
+2σ2
+v/τv
+(ω2 + ((2τv)−1 + ρ)2)(ω2 + ((2τv)−1 − ρ)2),
+(F13)
+where ρ =
+�
+(4τ 2v )−1 − ω2
+0. The chemotaxis signalling network is fully determined by the following matrices (Eq. A1)
+G = q
+�
+1/(ατm) 0
+−γ
+0
+�
+,
+(F14)
+J =
+�
+−1/τm
+0
+αγ
+−1/τr
+�
+,
+(F15)
+B =
+��
+⟨η2m⟩
+0
+0
+�
+⟨η2x⟩
+�
+.
+(F16)
+The Fourier transform of the matrix exponential of the Jacobian is
+F{eJ t} = (iωIn − J )−1
+=
+�
+�
+1
+1/τm+iω
+0
+αγ
+(1/τm+iω)(1/τr+iω)
+1
+1/τr+iω
+�
+� ,
+(F17)
+which allows us to determine the gain matrix via G(ω) = F{eJ t}(ω)G, and the noise matrix using N(ω) = F{eJ t}(ω)B;
+see also Eq. A4 and Eq. A5.
+To gain more insight in the way in which the network maps the signal onto its output, we first study the integration
+kernels of the system. The integration kernel from ligand concentration to output is given by the inverse Fourier
+transform of element (1, 2) of the gain matrix G(ω), which is
+k(t) ≡ F−1{˜gℓ→x(ω)} = κNf(1 − f)(1 − p)XT
+1
+1 − τr/τm
+� 1
+τm
+e−τ/τm − 1
+τr
+e−τ/τr
+�
+,
+(F18)
+with κ = (¯ℓ+KI
+D)−1−(¯ℓ+KA
+D)−1. Due to the adaptive nature of the network, the static gain from ligand concentration
+to output is zero: ¯gℓ→x =
+� ∞
+0
+k(t)dt = 0; the long-time response to a step change in a constant input is zero. The
+kernel does indeed not change the output based on the input concentration directly, but instead takes a (time-averaged)
+derivative of the input (Fig. 8A). It is therefore useful to consider the kernel that maps the signal derivative onto the
+output. This kernel can be found by rearranging the expression for the output of a linear signalling network, Eq. D1.
+Disregarding the noise terms and integrating by parts gives
+� 0
+−∞
+k(−t)ℓ(t)dt = K(−t)ℓ(t)|0
+−∞ −
+� 0
+−∞
+K(−t)v(t)dt,
+(F19)
+
+27
+where v(t) ≡ ˙ℓ and K(t) is the primitive of k(t). To make progress we first determine K(t),
+K(t) = κNf(1 − f)(1 − p)XT
+1
+1 − τr/τm
+�
+−e−τ/τm + e−τ/τr�
+.
+(F20)
+The form of K(t) is that of a simple exponential kernel with a delay (Fig. 8B). We thus have both K(0) = 0 and
+K(∞) = 0. It is now clear that the convolution over the ligand concentration simply maps onto the convolution over
+its derivative as
+� 0
+−∞
+k(−t)ℓ(t)dt = −
+� 0
+−∞
+K(−t)v(t)dt.
+(F21)
+The static gain of K(t) is ¯gv→x =
+� ∞
+0
+K(t)dt = qγτrτm = κNXT(1 − p)f(1 − f)τm. The gain thus increases with the
+number of receptors per cluster, N, the number of readout molecules, XT, and notably, with the adaptation time τm.
+This static gain from signal derivative to network output is a useful quantity which we will use to describe the other
+statistics of the network below.
+0
+5
+10
+15
+20
+-5
+0
+5
+10
+15
+20
+25
+30
+Time (s)
+-k(t): kernel ℓ → x*
+0
+5
+10
+15
+20
+-5
+0
+5
+10
+15
+20
+25
+30
+Time (s)
+K(t): kernel v → x*
+0.01
+0.10
+1
+10
+100
+0.01
+1
+100
+104
+Frequency ω (s-1)
+Power
+τr
+-1
+τm
+-1
+Nx
+2
+gℓ→x
+2
+gℓ→x
+2 /Nx
+2
+A
+B
+C
+FIG. 8. Integration kernel and power spectra. (A) The integration kernel k(t) takes a temporal derivative by weighing the
+most recent signal values with an opposite sign from the preceding ones. (B) The integration kernel K(t) from the derivative
+of the input concentration to the network output. The kernel K(t) is the primitive of k(t), and its static gain is proportional
+to the adaptation timescale τm. (C) Frequency dependent gain ˜g2
+ℓ→x(ω), frequency dependent noise N 2
+x(ω), and their ratio, as
+a function of frequency. The chemotaxis network is a band-pass filter, the frequencies that are passed through are set by τr
+on the high end and τm on the low end. At low frequencies, the methylation noise dominates. Parameters used in all panels
+τr = 0.1s and τm = 10s. Model parameters are ˜a = 2, N = 6, KI
+D = 18µM, KA
+D = 2900µM, ¯ℓ = 100µM, p = f = 0.5.
+To compute the past and predictive information, we need to determine the variance in the output, the SNR, and the
+correlation between the current output and the future ligand derivative. To that end we require the power spectrum
+of the output, and the cross-spectrum from output to future derivative. For the power spectrum of the output we use
+Eq. A5 to find
+Sx(ω) =
+q2γ2ω2
+(τr−2 + ω2)(τm−2 + ω2)Sℓ(ω) +
+α2γ2⟨η2
+m⟩
+(τr−2 + ω2)(τm−2 + ω2) +
+⟨η2
+m⟩
+τr−2 + ω2 .
+(F22)
+From this power spectrum we can see that the network is a band-pass filter, where the gain is maximal in the frequency
+range τm−1 < ω < τr−1. Both for ω ≫ τr−1 and ω ≪ τm−1 the gain goes to 0. On long timescales the methylation
+noise dominates (Fig. 8C). The cross-power spectrum between current output and future ligand derivative is given by
+element (2, 2) of the matrix G(−ω)Ss(ω) which is (also see Eq. A6)
+Sx→v(ω) = qγ
+−ω2Sℓ(ω)
+(τm−1 − iω)(τr−1 − iω).
+(F23)
+In the main text, we argue that the biologically relevant regime of the input signal is the limit ω0 → 0. We therefore
+present below the network statistics in this limit. We start by determining the variance in the readout, via the inverse
+Fourier transform of its power spectrum (Eq. F22):
+lim
+ω0→0 σ2
+x = ¯g2
+v→x
+1 + τr/τv + τr/τm
+(1 + τm/τv)(1 + τr/τv)(1 + τr/τm)σ2
+v + ¯g2
+a→xαRTp(1 − p)
+1
+1 + τr/τm
++ XTf(1 − f),
+= ¯g2
+v→x
+1 + τr/τv + τr/τm
+(1 + τm/τv)(1 + τr/τv)(1 + τr/τm)
+�
+��
+�
+dynamical gain
+σ2
+v + XTf(1 − f)
+�
+1 + ¯gv→x
+˜α(1 − p)
+RTκτm
+1
+1 + τr/τm
+�
+,
+(F24)
+
+28
+where ¯ga→x = γτr = XTf(1−f)/(RTp) is the static gain from receptor activity to readout, and we used the definition
+of α = ˜αNp(1 − p). Because there is no receptor-ligand binding noise, there is also no time averaging as in the
+push-pull network (and hence no factor depending on τr/τc). There is methylation noise on a timescale τm, but
+this cannot be time-averaged effectively because the integration time τr of the push-pull network is shorter than the
+receptor methylation timescale τm. The methylation noise can only be averaged out significantly by increasing RT.
+The contribution from the variance in the signal derivative, σ2
+v, to the output noise σ2
+x, depends on the dynamical gain,
+which is the product of the static gain ¯g2
+v→v and a factor that only depends on ratios of timescales. The dynamical
+gain is maximized for τr → 0 and τm → ∞, which is intuitive since subtracting a signal from an earlier one reduces
+the amplification of the signal. Hence, when the system has too few XT molecules to lift the signal above the noise,
+τm must be increased to raise the gain. Only when XT is sufficiently large, can τm be reduced. This allows the system
+to take more recent derivatives. The signal to noise ratio SNR = σ2
+x|η/σ2
+x|L can straightforwardly be obtained from
+Eq. F24. For the covariance between the current output and the future derivative we have
+lim
+ω0→0⟨δx(0)δv(τ)⟩ = F−1{Sx→v(ω)},
+=
+−¯gv→xσ2
+v
+(1 + τm/τv)(1 + τr/τv)e−τ/τv.
+(F25)
+The variance in Eq. F24 can be used to obtain the normalized correlation function ⟨δx(0)δv(τ)⟩/(σxσv).
+4.
+Past and predictive information of the chemotaxis network
+The past and predictive information are straightforward to compute from the quantities above. The definition of
+the past information is the same as for the push-pull network, and is given by Eq. D3. The SNR is now given by,
+using Eq. F24:
+SNR = σ2
+x|η/σ2
+x|L = κ2Nτ 2
+mσ2
+v
+1 + τr/τv + τr/τm
+(1 + τm/τv)(1 + τr/τv)(1 + τr/τm)
+� �
+1
+NXTf(1 − f)(1 − p)2 +
+˜α
+RT(1 + τr/τm)
+�
+,
+where κ = (¯ℓ + KI
+D)−1 − (¯ℓ + KA
+D)−1. The predictive information is found in the same manner as in Eq. D6, but now
+it is a function of the correlation between the current output and the future derivative of the ligand concentration.
+This correlation can be decomposed into the instantaneous correlation coefficient and an exponential decay on the
+timescale of the fluctuations of the derivative of the concentration, Eq. F25. Specifically, the predictive information
+is given by
+Ipred(x0; vτ) = −1
+2 log(1 − ρ2
+ℓve−2τ/τv).
+(F26)
+The instantaneous correlation coefficient ρ2
+ℓv can be found using Eq. F25 and Eq. F24. From Eq. F26 it is clear
+that just like for the push-pull network, the optimal design of the network that maximizes the predictive information,
+determined by the optimal ratio XT/RT and the optimal adaptation time τm, does not depend on the forecast interval
+τ. The forecast interval only affects the magnitude of the predictive information.
+5.
+Optimal allocation
+We can determine the optimal ratio (XT/RT)opt that maximizes either the past information or the predictive infor-
+mation, given all other network parameters, most notably τm. Just as for the push-pull network, we find however that
+the optimal ratio (XT/RT)opt is the same regardless of whether the past or the predictive information is maximized.
+This is again because the information on the future signal (be it the value or the derivative) is encoded in the receptor
+occupancy, while the ratio XT/RT controls the interval by which the downstream readout samples the receptor to
+estimates its occupancy. Nonetheless, the optimal methylation timescale τmopt that maximizes either the past or the
+predictive information is different—maximizing predictive information requires a more recent derivative and hence a
+shorter τm than obtaining past information.
+Given τm and all other parameters, the optimal ratio of the number of readout molecules over receptor clusters is,
+
+29
+using C = RT + XT,
+�XT
+RT
+�opt
+=
+�
+1
+α
+1
+f(1 − f)
+p
+1 − p
+�
+1 + τr
+τm
+,
+= 2
+�
+2/N
+�
+1 + τr
+τm
+,
+(F27)
+where in the second line we have used that α = ˜αNp(1 − p), and ˜α = 2, and f = p = 0.5. Because for the chemotaxis
+network τr < τm the ratio τr/τm only varies between 0 and 1. For this reason, the optimal ratio (XT/RT)opt depends
+only weakly on τm, and does not vary strongly along the isocost lines of Fig. 4A in the main text, see Fig. 9.
+0.00
+0.05
+0.10
+0.15
+0.20
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Distance along fixed cost line θ
+XT/RT
+FIG. 9. The optimal allocation ratio XT/RT varies only slightly along the isocost lines of Fig. 4A in the main
+text. The optimal ratio XT/RT as a function of the distance θ along the isocost lines of Fig. 4A of the main text; dotted line
+C = 102, solid line C = 104, dashed line C = 106. The red dots mark the points where the predictive information is maximal.
+Along the isocost lines XT/RT varies much more weakly than for the push-pull network; for resource availability C ≤ 104 the
+ratio is almost constant. Parameters used g = 4mm−1, τr = 0.1s, KI
+D = 18µM, KA
+D = 2900µM, N = 6, ˜α = 2, p = f = 0.5,
+¯ℓ = 100µM.
+
+30
+[1] M. Monti, D. K. Lubensky, and P. R. ten Wolde, Robust-
+ness of Clocks to Input Noise, Physical Review Letters
+121, 078101 (2018), publisher: American Physical Soci-
+ety.
+[2] W. Pittayakanchit, Z. Lu, J. Chew, M. J. Rust, and
+A. Murugan, Biophysical clocks face a trade-off between
+internal and external noise resistance., eLife 7 (2018).
+[3] E. Kussell and S. Leibler, Phenotypic Diversity, Pop-
+ulation Growth, and Information in Fluctuating Envi-
+ronments, Science 10.1126/science.1114383 (2005), pub-
+lisher: American Association for the Advancement of Sci-
+ence.
+[4] I. Tagkopoulos, Y.-C. Liu, and S. Tavazoie, Predic-
+tive Behavior Within Microbial Genetic Networks, Sci-
+ence 10.1126/science.1154456 (2008), publisher: Ameri-
+can Association for the Advancement of Science.
+[5] A. Mitchell, G. H. Romano, B. Groisman, A. Yona,
+E. Dekel, M. Kupiec, O. Dahan, and Y. Pilpel, Adaptive
+prediction of environmental changes by microorganisms,
+Nature 460, 220 (2009), number: 7252 Publisher: Nature
+Publishing Group.
+[6] W. Bialek, Biophysics: searching for principles, edited by
+Princeton Editorial Associates Inc. (Princeton University
+Press, Woodstock, Oxfordshire, 2012) pages: 563.
+[7] N. B. Becker, A. Mugler, and P. R. Ten Wolde, Opti-
+mal Prediction by Cellular Signaling Networks, Physical
+Review Letters 115, 1 (2015).
+[8] N. Tishby, F. C. Pereira, and W. Bialek, The information
+bottleneck method, Proceedings of 37th Allerton Confer-
+ence on communication and computation (1999).
+[9] V. Sachdeva, T. Mora, A. M. Walczak, and S. E. Palmer,
+Optimal prediction with resource constraints using the
+information bottleneck, PLOS Computational Biology
+17, e1008743 (2021), publisher: Public Library of Sci-
+ence.
+[10] C. C. Govern and P. R. ten Wolde, Optimal resource
+allocation in cellular sensing systems, Proceedings of the
+National Academy of Sciences 111, 17486 LP (2014).
+[11] G. Malaguti and P. R. ten Wolde, Theory for the opti-
+mal detection of time-varying signals in cellular sensing
+systems, eLife 10, e62574 (2021).
+[12] G. Chechik, A. Globerson, N. Tishby, and Y. Weiss, In-
+formation Bottleneck for Gaussian Variables, Journal of
+Machine Learning Research 6, 165 (2005).
+[13] A. Goldbeter and D. E. Koshland, An amplified sensitiv-
+ity arising from covalent modification in biological sys-
+tems, Proceedings of the National Academy of Sciences
+78, 6840 (1981), publisher: National Academy of Sci-
+ences Section: Research Article.
+[14] U. Alon, An Introduction to Systems Biology:
+De-
+sign Principles of Biological Circuits (Chapman and
+Hall/CRC, New York, 2006).
+[15] J. E. Segall, S. M. Block, and H. C. Berg, Temporal com-
+parisons in bacterial chemotaxis., Proceedings of the Na-
+tional Academy of Sciences of the United States of Amer-
+ica 83, 8987 (1986), publisher: Proc Natl Acad Sci U S
+A.
+[16] C. C. Govern and P. R. ten Wolde, Optimal resource
+allocation in cellular sensing systems, Proceedings of the
+National Academy of Sciences of the United States of
+America 111, 17486 (2014).
+[17] M. Hinczewski and D. Thirumalai, Cellular Signaling
+Networks Function as Generalized Wiener-Kolmogorov
+Filters to Suppress Noise, Physical Review X 4, 3 (2014).
+[18] H. H. Mattingly,
+K. Kamino,
+B. B. Machta, and
+T. Emonet, Escherichia coli chemotaxis is information
+limited, Nature Physics 2021 17:12 17, 1426 (2021), pub-
+lisher: Nature Publishing Group.
+[19] T.-L.
+Wang,
+B.
+Kuznets-Speck,
+J.
+Broderick,
+and
+M. Hinczewski, The price of a bit:
+energetic costs
+and
+the
+evolution
+of
+cellular
+signaling,
+bioRxiv
+,
+2020.10.06.327700 (2022).
+[20] S. Tanase-Nicola, P. B. Warren, and P. R. ten Wolde, Sig-
+nal detection, modularity, and the correlation between
+extrinsic and intrinsic noise in biochemical networks,
+Phys. Rev. Lett. 97, 068102 (2006).
+[21] E. Ziv, I. Nemenman, and C. H. Wiggins, Optimal sig-
+nal processing in small stochastic biochemical networks,
+PLoS One 2, 10.1371 (2007).
+[22] W. De Ronde, F. Tostevin, and P. R. ten Wolde, Effect
+of feedback on the fidelity of information transmission of
+time-varying signals, Phys. Rev. E 82, 031914 (2010).
+[23] T. E. Ouldridge, C. C. Govern, and P. R. ten Wolde,
+Thermodynamics of computational copying in biochem-
+ical
+systems,
+Physical
+Review
+X
+7,
+10.1103/Phys-
+RevX.7.021004 (2017).
+[24] J. E. Keymer, R. G. Endres, M. Skoge, Y. Meir, and
+N. S. Wingreen, Chemosensing in Escherichia coli: Two
+regimes of two-state receptors, Proceedings of the Na-
+tional Academy of Sciences 103, 1786 (2006), publisher:
+National Academy of Sciences.
+[25] N. Barkai and S. Leibler, Robustness in simple biochem-
+ical networks, Nature 387, 913 (1997).
+[26] R. G. Endres and N. S. Wingreen, Precise adaptation
+in bacterial chemotaxis through “assistance neighbor-
+hoods”, Proceedings of the National Academy of Sciences
+103, 13040 (2006).
+[27] G. Lan, P. Sartori, S. Neumann, V. Sourjik, and Y. Tu,
+The energy–speed–accuracy trade-off in sensory adapta-
+tion, Nature Physics 8, 422 (2012).
+[28] P. Sartori and Y. Tu, Free Energy Cost of Reducing Noise
+while Maintaining a High Sensitivity, Physical Review
+Letters 115, 118102 (2015), 1505.07413.
+[29] T. S. Shimizu, Y. Tu, and H. C. Berg, A modular
+gradient-sensing network for chemotaxis in Escherichia
+coli revealed by responses to time-varying stimuli, Molec-
+ular Systems Biology 6, 1 (2010), publisher: Nature Pub-
+lishing Group.
+[30] Y. Tu, T. S. Shimizu, and H. C. Berg, Modeling the
+chemotactic response of Escherichia coli to time-varying
+stimuli, Proceedings of the National Academy of Sciences
+of the United States of America 105, 14855 (2008).
+[31] F. Tostevin and P. R. Ten Wolde, Mutual information be-
+tween input and output trajectories of biochemical net-
+works, Physical Review Letters 102, 1 (2009).
+[32] M. Reinhardt, G. Tkaˇcik, and P. R. t. Wolde, Path
+Weight Sampling: Exact Monte Carlo Computation of
+the Mutual Information between Stochastic Trajectories,
+arXiv 10.48550/arxiv.2203.03461 (2022), 2203.03461.
+[33] J. Monod, J. Wyman, and J.-P. Changeux, On the nature
+of allosteric transitions: A plausible model, Journal of
+Molecular Biology 12, 88 (1965).
+
+31
+[34] V. Sourjik and H. C. Berg, Binding of the Escherichia coli
+response regulator CheY to its target measured in vivo
+by fluorescence resonance energy transfer, Proceedings of
+the National Academy of Sciences of the United States of
+America 99, 12669 (2002), publisher: National Academy
+of Sciences.
+[35] M. Li and G. L. Hazelbauer, Cellular stoichiometry of the
+components of the chemotaxis signaling complex, Journal
+of Bacteriology 186, 3687 (2004), publisher: J Bacteriol.
+[36] G. Malaguti and P. R. T. Wolde, Theory for the optimal
+detection of time-varying signals in cellular sensing sys-
+tems, eLife 10, 1 (2021), publisher: eLife Sciences Pub-
+lications Ltd.
+[37] P. Sartori and Y. Tu, Free Energy Cost of Reducing Noise
+while Maintaining a High Sensitivity, Physical Review
+Letters 115, 118102 (2015), publisher: American Physi-
+cal Society.
+[38] J. O. Dubuis, G. Tkacik, E. F. Wieschaus, T. Gregor, and
+W. Bialek, Positional information, in bits, Proceedings of
+the National Academy of Sciences of the United States
+of America 110, 16301 (2013).
+[39] N. Van Kampen, Stochastic processes in physics and
+chemistry (North Holland, Amsterdam, 1992).
+[40] E. Ziv, I. Nemenman, and C. H. Wiggins, Optimal sig-
+nal processing in small stochastic biochemical networks.,
+PLoS ONE 2, e1077 (2007).
+[41] M. Vennettilli, S. Saha, U. Roy, and A. Mugler, Precision
+of Protein Thermometry, Physical Review Letters 127,
+098102 (2021), publisher: American Physical Society.
+[42] V. Sourjik and H. C. Berg, Functional interactions be-
+tween receptors in bacterial chemotaxis, Nature 2004
+428:6981 428, 437 (2004), publisher: Nature Publishing
+Group.
+[43] B. A. Mello and Y. Tu, Effects of adaptation in main-
+taining high sensitivity over a wide range of backgrounds
+for Escherichia coli chemotaxis, Biophysical Journal 92,
+2329 (2007), publisher: Biophysical Society.
+[44] J. R. Maddock and L. Shapiro, Polar Location of the
+Chemoreceptor Complex in the Escherichia coli Cell, Sci-
+ence 259, 1717 (1993), publisher: American Association
+for the Advancement of Science.
+[45] T. A. J. Duke and D. Bray, Heightened sensitivity of a
+lattice of membrane receptors, Proceedings of the Na-
+tional Academy of Sciences 96, 10104 (1999).
+[46] J. M. Keegstra, K. Kamino, F. Anquez, M. D. Lazova,
+T. Emonet, and T. S. Shimizu, Phenotypic diversity and
+temporal variability in a bacterial signaling network re-
+vealed by single-cell FRET., eLife 6, 708 (2017).
+[47] J. S. Parkinson, G. L. Hazelbauer, and J. J. Falke, Signal-
+ing and sensory adaptation in Escherichia coli chemore-
+ceptors: 2015 update, Trends in Microbiology Special Is-
+sue: Microbial Translocation, 23, 257 (2015).
+[48] B. A. Mello and Y. Tu, An allosteric model for het-
+erogeneous receptor complexes:
+Understanding bacte-
+rial chemotaxis responses to multiple stimuli, Proceed-
+ings of the National Academy of Sciences of the United
+States of America 102, 17354 (2005), publisher: National
+Academy of Sciences.
+[49] K. Kamino, J. M. Keegstra, J. Long, T. Emonet, and
+T. S. Shimizu, Adaptive tuning of cell sensory diversity
+without changes in gene expression, Science Advances
+6, eabc1087 (2020), publisher: American Association for
+the Advancement of Science.
+[50] T. Emonet and P. Cluzel, Relationship between cellular
+response and behavioral variability in bacterial chemo-
+taxis, Proceedings of the National Academy of Sciences
+105, 3304 (2008), 0705.4635.
+[51] M. N. Levit, T. W. Grebe, and J. B. Stock, Organiza-
+tion of the Receptor-Kinase Signaling Array That Regu-
+lates Escherichia coli Chemotaxis *, Journal of Biological
+Chemistry 277, 36748 (2002), publisher: Elsevier.
+