diff --git "a/6dAzT4oBgHgl3EQff_zo/content/tmp_files/load_file.txt" "b/6dAzT4oBgHgl3EQff_zo/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/6dAzT4oBgHgl3EQff_zo/content/tmp_files/load_file.txt" @@ -0,0 +1,559 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf,len=558 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='01463v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='soft] 4 Jan 2023 Mechanosensitive bonds induced complex cell motility patterns Jen-Yu Lo1, Yuan-Heng Tseng1 and Hsuan-Yi Chen 1,2,3 1Department of Physics, National Central University, Jhongli 32001, Taiwan 2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan 3Physics Division, National Central for Theoretical Sciences, Taipei, 10617, Taiwan (Dated: January 5, 2023) The one-dimensional crawling movement of a cell is considered in this theoretical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our active gel model shows that for a cell with weakly mechanosensitive adhesion complexes, as myosin contrac- tility increases, a cell starts to move at a constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As the mechanosensitivity of the adhesion complexes increases, a cell can exhibit stick-slip motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, a cell with highly mechanosensitive adhesion complexes exhibits periodic back-and-forth migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' A simplified model which assumes that the cell crawling dynamics are controlled by the evolution of the myosin density dipole and the asymmetry of adhesion complex distribution captures the motility behaviors of crawling cells qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It suggests that the complex cell crawling behaviors observed in the experiments could result from the interplay between the distribution of contractile force and mechanosensitive bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' PACS numbers: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='Jj,87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='Uv Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– The crawling motion of eukaryotic cells is ubiquitous in biology as it plays important roles in processes such as embryogenesis, wound healing, cancer metastasis, and immunology [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Common if not uni- versal features of a crawling cell include myosin motors distributed mainly behind the center, dominant actin polymerization in the leading edge, and higher density of adhesion complexes in the leading region [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Such po- larized molecular distribution enables protrusion in the leading edge due to actin polymerization, treadmilling of actomyosin cytoskeleton due to contractility, and traction force pulling the cell body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' These features, therefore, are included in many theoretical models for crawling cells [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Interestingly, besides non-motile resting and steady- moving behaviors, cells crawling along a one-dimensional track either on a substrate or in a three-dimensional en- vironment also exhibit moving patterns that are non- stationary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For example, stick-slip crawling mo- tion due to slip between integrin and the extracellular matrix in focal adhesions under the contractility provided by myosin II has been observed in human osteosarcoma cells [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Periodic back-and-forth migration has been ob- served in crawling zyxin-depleted cells in a collagen ma- trix [5] and dendritic cells crawling along microfabricated channels [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Several theoretical models have been proposed to ex- plain some of these deterministic complex moving pat- terns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' A model that includes the mechanochemical cou- pling of actin promotor dynamics and actin polymer- ization to myosin kinetics was shown to produce peri- odic back-and-forth migration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, a purely mechanical model emphasizing the interplay be- tween mechanosensitive bonds and membrane tension ex- hibited stick-slip motion even for slip bonds [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Inter- estingly, it has also been shown that stick-slip can re- sult from the interplay between mechanosensitive bonds, contractility, and a force that tends to restore a cell’s preferred length [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this letter, we present a theoretical study to show that the coupling between mechanosensitive adhesion complexes and myosin contractility is sufficient to gener- ate deterministic complex cell crawling behaviors, includ- ing stick-slip, periodic back-and-forth movements, and other complex moving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We first construct an active gel model with mechanosensitive adhesion com- plexes and show that the distribution of myosin motors and adhesion complexes computed from our model agree with experimentally observed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' By exploring the motility behavior with different strengths of contractil- ity and mechanosensitivity, we show that this model can lead to complex motility behaviors other than rest and constant-velocity moving states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Among these complex motility patterns, unidirectional stick-slip motion and pe- riodic back-and-forth movement are the most common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' When the adhesion complexes are less mechanosensitive, as myosin contractility increases, the cell performs con- stant velocity motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, for cells with highly mechanosensitive adhesion complexes, as myosin contractility becomes sufficiently strong, periodic back- and-forth crawling motion can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, stick-slip and other complex motility patterns can be ob- served by increasing the mechanosensitivity of the adhe- sion complexes for a cell moving at constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' To understand the physical mechanisms that produce these complex motility patterns, a simplified model in- spired by the active gel model is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The sim- plified model assumes that the dynamics of a sufficiently slow-moving cell are dominated by the dipole moment of myosin density and the difference in the total num- ber of adhesion complexes near the two cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Re- markably, the motility behavior predicted by this simple model agrees qualitatively with the motility behaviors predicted by the active gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' These results sug- gest that, in general, diverse complex motility behaviors can result from the interplay between mechanosensitive adhesions and the dynamical organization of contractile 2 myosin motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Model summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– To focus on the contribution of cell mechanics to motility behaviors, chemical signaling is not included in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The cytoplasm of the cell is mod- eled as an active gel [10, 11] enclosed by the cell mem- brane, the adhesion complexes are treated as reversible bonds with specific binding-unbinding rates, and actin polymerization is assumed to happen only at the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The forces acting on the cell include the stress in the cytoskeleton, the drag force from the substrate, and the force due to the adhesion complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our model only considers one spatial direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', the cell’s moving direction, and the stress in the cytoplasm obeys the constitutive equation σ = η ∂v ∂x + χc, (1) where η is the effective one-dimensional viscosity of the cytoplasm, v is the flow field, χ is the strength of contrac- tility provided by myosin motors (χ > 0), and c is the concentration of myosin attached to the actin network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For simplicity, compressibility is not included [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Thus pressure does not appear in the constitutive relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The force exerted by the substrate is Fdrag = −αnbv − ξv, (2) The first term on the right hand is the drag provided by the adhesion complexes [12], α is a constant that char- acterizes the resistance of the adhesion complexes to cell movement, and nb is the number density of adhesion com- plexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The second term comes from the viscous drag of the fluid between the cell and the substrate, and ξ is the drag coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Putting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (1)(2) together, the result- ing force balance equation, ∂xσ + Fdrag = 0, takes the following form η ∂2v ∂x2 − (αnb + ξ)v = −χ ∂c ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (3) Myosin motors attached to actin filaments move with the cytoplasm, while those detached from actin filaments diffuse freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The attachment/detachment of motors is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On long-time scales, the density of the mo- tors can be effectively described by an advection-diffusion equation [13] ∂c ∂t = D ∂2c ∂x2 − ∂(cv) ∂x , (4) where D is the effective diffusion coefficient of myosin motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Adhesion complexes providing anchorage to the extra- cellular matrix are also physically coupled to the con- tractile cytoplasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As a result, they are pulled when the cytoplasm moves [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Once an adhesion complex is formed, the adhesion site does not move, but the disso- ciation rate of the adhesion complex is affected by the motion of the cytoskeleton because the bond is stretched or compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In our model, the evolution of the density of adhesion complexes is assumed to obey ∂nb ∂t = −k0 e−k1∂xvnb + kon, (5) where kon is the binding rate, k0 is the unbinding rate at ∂xv = 0, and k1 tells us how unbinding rate is affected by the cytoplasmic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our model assumes that when the strain rate is dilating, giving more space for the adhesion complexes, the unbinding rate decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Actin polymerization at the cell ends depends on the distribution of actin activators [15][16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In the presence of environmental cues, a gradient of actin activator con- centration within the cell is established, and actin poly- merization is polarized due to this concentration gradi- ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In the absence of such external influence, the cell can nevertheless polarize itself by spontaneous symmetry breaking, and the net actin polymerization rate becomes asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In our model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' we consider a homogeneous environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' and the net actin polymerization rate v± p at the ± end of the cell is assumed to be v± p = 2 e−v(1) p (L−L0) 1 + exp[∓ dl± dt /v(2) p ] v(0) p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (6) where v+ p (v− p ) is the net rate of extension due to actin polymerization at the cell end located at x = l+(l−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' v(0) p comes from the base polymerization rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' v(1) p in the exponent of the numerator comes from the effect of free energy cost for polymerization when the cell length is different from its natural length (L0 is the natural length of the cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' and L = l+ − l−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' and the term with v(2) p makes the net polymerization rate in a moving cell at both ends different,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' with more polymerization events in the leading end than the trailing end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It will become clear that the qualitative results of cell motility behavior do not depend on the specific form we assumed for the dissociation rate of the adhesion complexes and v± p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The evolution of cell-end positions is determined by the velocity of cytoplasm and actin polymerization, dl± dt = v± ± v± p , (7) where v+ (v−) is the velocity of cytoplasm at the + (−) end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Experimentally it has been shown that a cell tends to restore its length L to a preferred magnitude L0 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We model this effect by the following force balance condition at cell ends σ± = � χc + η ∂v ∂x � l± = −γ(L − L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (8) Here γ is a constant associated with the restoring force that brings the cell length L to L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3 Because no myosin motors can leave or enter the cell, the total flux of myosin motors across a cell end should vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This leads to [cv]l± − [c]l± dl± dt − D � ∂c ∂x � l± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (9) The first two terms on the left-hand side are the advective flux relative to the moving cell end, and the last term is the diffusive flux at the cell end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is convenient to introduce effective drag coefficient ξeff = ξ + αkon/k0 and choose l0 = � η/ξeff as the unit length, t0 = η/(ξeffD) as the unit time, σ0 = ξeffD as the unit stress, n0 = ξ/α as the unit density for adhe- sion complexes, and c0 = M/ � η/ξeff as the unit myosin concentration, where M is the total number of myosin motors in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore the dimensionless drag co- efficient ˜ξ = ξ/(ξ + αkon/k0), contractility ˜χ = c0χ/σ0, and cell elastic constant K = γl0/σ0 are used in the fol- lowing discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Simulation of motility behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– From the point of view of nonequilibrium thermodynamics, the drag force between the cell and the substrate, the viscous force in the cytoplasm, and the diffusion of myosin are passive processes against cell movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, ac- tive processes such as actin polymerization and myosin contractility drive the movement of the cell, and the binding/unbinding dynamics of the adhesion complexes modulate cell movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As the myosin motors pro- vide contractility against viscous and substrate drag, the contractility-induced cytoplasmic flow drifts the motors to aggregate and also affects the distribution of adhesion complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Once the cell is in motion, the feedback in the actin polymerization rate further enhances cell polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The balance between these processes determines the state of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In general, there is no analytical so- lution when all these effects are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore we numerically integrate the equations of motion by a finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The details of our numerical methods and our choice of parameters are discussed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure 1 shows the motility behaviors for a cell with parameters chosen to be compatible with typical cells [20] and a range of adhesion complex mechanosensitivity and contractility strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The following motility behaviors are found: rest, moving at a constant velocity, unidirec- tional stick-slip movement, back-and-forth motion with stick-slip, and periodic back-and-forth movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For a cell with weakly mechanosensitive adhesion complexes, as contractility increases, a cell at rest starts to move at con- stant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As the adhesion complexes become more mechanosensitive, a moving cell shows other complex motility behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For example, stick-slip motion and (at high contractivity) back-and-forth motion with stick- slip, and finally, the cell performs periodic back-and- forth motion when the adhesion complexes are highly mechanosensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Another motility phase diagram in the Supplement Materials [20] shows that, within our model, a cell with a high actin polymerization rate can exhibit other complex motility behaviors between stick-slip and periodic back-and-forth movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' �� �� �� �� �� �� � � ��� ��� ��� ��� � � (a) � � � � � � �� � (b) � � �� � � �� �� � (c) � �� �� � ��� ��� ���� � (d) � � �� � ��� ��� ��� � (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (a) Motility phase diagram for a cell with dimension- less parameters K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, and v(2) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Rest state (squares), constant- velocity motion (diamonds), stick-slip movement (triangles), back-and-forth with stick-slip motion (empty circle), and pe- riodic back-and-forth motion (filled circles) are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (b) Trajectories of the cell ends for ˜χ = 18, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='05, the cell performs constant velocity motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (c) Trajectories of the cell ends for ˜χ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, the cell performs stick-slip mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (d) Trajectories of the cell ends for ˜χ = 19, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='15, the cell performs complex motility pattern which is periodic back-and-forth with stick-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (e) Trajectories of the cell ends for ˜χ = 18, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='25, the cell performs periodic back-and- forth motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure 2 shows that when the cell is at rest, myosin motor distribution is symmetric around the center of the cell, and the number of adhesion complexes near both cell ends is the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' on the other hand, for a cell mov- ing at constant velocity, myosin motors aggregate close to the trailing end and adhesion complexes are mainly close to the leading end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The distribution of myosin mo- tors and adhesion complexes for a cell undergoes stick- slip, and periodic back-and-forth movements are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S2 of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is clear that whenever the cell has a definite moving direction, myosin motors aggregate in a regime behind the center of the cell, and more adhesion complexes form near the leading end than the trailing end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Indeed, the adhesion complex binding/unbinding rates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (5) and net actin polymerization at the cell ends Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (6) in our model lead to reasonable molecular distributions in a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Reduction to the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– To obtain an in- 4 ���� ��� ��� � ��� ��� ��� � � � (a) ���� ��� ��� � ��� ��� ��� (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Distribution of myosin motors and adhesion com- plexes for a cell (a) at rest and (b) undergoes constant velocity motion towards +x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The parameters are the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 1 and (a) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='25, ˜χ = 16, (b) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, ˜χ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' tuitive physical picture of the complex motility behav- iors, especially the transitions from the rest state to the constant-velocity movement and periodic back-and-forth movement, we construct a simplified model from the ac- tive gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, we consider a limiting situation in which adhesion complexes only appear in a small region close to the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this regime, it is convenient to in- troduce Nf(Nb), the total number of adhesion complexes close to l+(l−), and N = Nf + Nb, ∆N = Nf − Nb to describe the distribution of adhesion complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We also introduce yc, the dipole moment of myosin motors den- sity relative to the center of the cell [20], to characterize the spatial distribution of myosin motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The net actin polymerization velocity at the cell ends v± p ≡ vp±∆vp/2, where ∆vp = v+ p − v− p ∝ Vcell in the limit of small cell velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore we write v± p = vp ± βVcell/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this regime, straightforward calculation shows that the veloc- ity of the cell is [20] Vcell ≈ 1 1 − β/2(λν1vp∆N − ˜χλν2yc), (10) where λν1 and λν2 are positivet coefficients that depend on N and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Note that, from the definition, β/2 cannot be greater than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Therefore a cell with ∆N > 0 and yc < 0 has positive Vcell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is in agreement with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' We further simplify the analysis by considering the limit of large K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', L ≈ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The following equations are constructed to describe the dynamics of N, ∆N, and yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, the evolution equations for N and ∆N are dN dt = 2kon − k(0) off N − k(1) off yc ∆N, d∆N dt = −k(0) off ∆N − k(1) off Nyc, (11) where kon, k(0) off , and k(1) off are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Next, in the spirit of Landau-type approximation, the evolution of yc is assumed to obey the following equation, dyc dt = −Γ � −(˜χ − ˜χc)yc − a∆N∆N + a3y3 c � , (12) where ˜χc and a∆N are treated as N-independent con- stants for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The simple model equations (11)(12) have a solution with constant N and ∆N = yc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This corresponds to a cell at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Solutions with nonzero constant ∆N and yc correspond to a cell moving at a constant veloc- ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' solutions with time-periodic ∆N and yc are stick-slip (periodic back-and-forth) movement if the time-average of ∆N and yc are nonzero (zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The linear stability analysis of the rest state shows that as the contractil- ity increases, a cell at rest starts to move as the system undergoes a bifurcation, the moving state is the constant- velocity state when k(1) off is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', the adhesion com- plexes are less mechanosensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' When k(1) off is sufficiently large, the rest-to-moving transition leads to a periodic back-and-forth moving state [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3 [20], the model Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (11) (12) exhibit a motility phase diagram qualitatively the same as the numerical solutions of our active gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The minor differences come from those simplifications made when constructing the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, from the simplified model, it is easy to see how the sym- metry properties and the couplings of the key driving variables lead to the observed cell motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For exam- ple, yc(t) and ∆N(t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3(a) for a cell performing periodic back-and-forth movement suggests the following physical picture about how the coupling terms in the sim- plified model lead to this motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (11), a small ∆N tend to increase when yc is sufficiently nega- tive, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (12) states that when a∆N > 0, yc tends to move toward the center of the cell when the magnitude of ∆N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The result is that at sufficiently large k(1) off , the number of adhesion complexes in the lead- ing end of a moving cell increases sufficiently fast such that at some point, the myosins are pulled to the other half of the cell, reversing the sign of yc, then reversing the sign of ∆N, and eventually the direction of cell motion is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is how rest/constant-velocity transition becomes rest/back-and-forth transition as the adhesion complexes are more mechanosensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In between the constant velocity and periodic back-and-forth movement, complex motility patterns with stick-slip can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3(b), yc(t) and ∆N(t) in a cell that un- dergoes stick-slip movement have nonzero time-average values, and they oscillate with similar phase-relations as a cell undergoes periodic back-and-forth movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is because the mechanosensitivity of the adhesion com- plexes is sufficiently strong to induce an oscillation of ∆N and yc, but not sufficiently strong to change their signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure 3(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3(d) show that yc(t) and ∆N(t) (defined as the difference of the total number of adhe- sion complexes in the leading and trailing halves of the cell) in the numerical simulations of the active gel model behave similarly, suggesting that the physical picture ob- tained from studying the simplified model can be applied 5 to more detailed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' � �� �� � ���� ��� ��� � � �� (a) � �� �� � ���� ��� ��� (b) ��� ��� ��� � ���� ��� ��� (c) � � �� � ���� ��� ��� (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (a)(b): yc(t) and ∆N(t) in the simplified model with kon/k(0) off = 1, a∆N = 1, and a3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In (a), k(1) off /k(0) off = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='6, Γ(˜χ− ˜χc)/k(0) off = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' in (b), k(1) off /k(0) off = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='62, Γ(˜χ− ˜χc)/k(0) off = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (c)(d): yc(t) and ∆N(t) in the active gel model with K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In (c), k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='55, ˜χ = 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' in (d), k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, ˜χ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The cell in (a)(c) performs oscillatory back-and-forth movement, and the cell in (b)(d) performs stick-slip movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='– Within our active gel model, a cell with highly mechanosensitive adhesion complexes can exhibit periodic back-and-forth movement similar to what was observed in zyxin-depleted cells in a collagen matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Since zyxin proteins act as mechanosensors in mature adhesion complexes [21], our study suggests that the dif- ference in the mechanosensitivity of the adhesion com- plexes in zyxin-depleted and wild-type cells could be the origin of the periodic back-and-forth movement observed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Future experiments can be designed to examine this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The physical picture suggested by our simplified model also implies the possibility that, in general, when some of the simplifications are lifted, more complex one- dimensional cell motility behaviors can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is indeed the case, as we explore the behavior of our ac- tive gel model for a broader range of actin polymerization rates, complex trajectories which come from further bi- furcations are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S1 in [20] and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Further study of the physical mechanisms for these behaviors will be our future work [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, although the physical mechanisms for symmetry-breaking transitions, such as rest/periodic back-and-forth transition and rest/constant velocity transition, can be understood from the dynamics of yc and ∆N, it is interesting to study how other important physical observables, such as the multipoles of the trac- tion force [23][24], behave in cells with different moving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is also important to check if the basic fea- tures of these physical observables in different moving patterns depend on the details of the binding/unbinding dynamics of adhesion complexes, as it plays a significant role in our understanding of many interesting features of cell motility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' � � �� � � � � � (a) � � �� � ��� ��� ��� � (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Our active gel model predicts that at high actin polymerization rates further complex motility behaviors, such as (a) zig-zag with stick-slip, and (b) double-period back-and- forth movement can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' These trajectorjes are obtained for K = 100, ˜χ = 14, kon = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='0, k0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='0, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In (a), v(0) p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' in (b), v(0) p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='1, k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The blue curves represent the trajectories of the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Acknowledgments – H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' thanks Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Jasnow (University of Pitts- burgh) for stimulating discussions and encouragement in the early stage of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' is supported by the Ministry of Science and Technology, Taiwan (MOST 108-2112-M-008-016 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The authors also acknowledge the support from National Center for Theoretical Sciences, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Bray, Cell Movements, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} 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free- boundary model of a motile cell explains turning behav- ior, PLoS Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Biol, 13, e1005862 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Horton and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Vandewalle, A space-time multigrid method for parabolic partial differential equations, SIAM Journal on Scientific Computing, 16, 848 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [20] See our Supplementary Material.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Ingber, Mechanical forces alter zyxin unbind- ing kinetics within focal adhesions of living cells, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Cell Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', 207, 187 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Lo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Chen, manuscript in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Tanimoto and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Sano, A simple force-motion rela- tion for migrating cells revealed by multipole analysis of traction stress, Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', 106, 16 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Ohta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Tarama, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' SanoA simple model of cell crawling, Physica D, 318, 3 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='01463v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='soft] 4 Jan 2023 Supplementary material for “ Mechanosensitive bonds induced complex cell motility patterns” Jen-Yu Lo1, Yuan-Heng Tseng1 and Hsuan-Yi Chen 1,2,3 1Department of Physics, National Central University, Jhongli 32001, Taiwan 2Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan 3Physics Division, National Central for Theoretical Sciences, Taipei, 10617, Taiwan (Dated: January 5, 2023) S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' DIMENSIONLESS EQUATIONS We introduce effective drag coefficient ξeff = ξ + αkon/k0 and choose l0 = � η/ξeff as the unit length, t0 = η/(ξeffD) as the unit time, σ0 = ξeffD as the unit stress, n0 = ξ/α (notice that, here ξ, not ξeff is used) as the unit density for adhesion complexes, and c0 = M/ � η/ξeff as the unit myosin concentration, where M is the total number of myosin motors in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In the dimensionless form, the momentum equation is ∂2v ∂x2 − ˜ξ(1 + nb)v = −˜χ ∂c ∂x, (S1) myosin density evolution obeys ∂c ∂t = ∂2c ∂x2 − ∂(cv) ∂x , (S2) evolution of the density of the adhesion complexes is ∂nb ∂t = −¯k0e−¯k1∂xvnb + ¯kon (S3) and the positions of the cell ends obey dl± dt = [v]l± ± v± p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S4) The boundary condition for myosin current is � c(v − dl dt) − ∂c ∂x � l± = 0, (S5) and the stress continuity at the cell ends leads to σ± = −K (L − L0) = �∂v ∂x � l± + ˜χcl±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S6) Here all variables and x, t are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The definitions of the parameters and important physical quantities are listed in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The dimensionless parameters in our model are listed in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' NUMERICAL METHOD AND CHOICE OF PARAMETERS In our numerical scheme, each iteration updates all dynamical variables by integrating the evolution equations over a small time interval ∆t with a finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, [v]l± and v± p from the previous iteration were substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S4) to obtain the new positions of the cell ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The densities of the adhesion complexes and myosin motors are updated from the flow field of the previous iteration by integrating the evolution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S3) of nb and the myosin advection-diffusion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The force balance Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S1) with the updated bond density and myosin concentration is then solved to obtain the new flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The numerics were carried out by dividing the cell into Nx = 100 segments and approximating the spatial derivatives by the finite difference method with the size of a time step ∆t = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The typical material parameters are D ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='025 µm2/s [1], l0 ∼ 10 µm and unit time t0 ∼ 103 s [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' k0 is the rate of dissociation of mature focal adhesion and S2 Physical meaning Symbol effective drag coefficient ξeff = ξ + αkon/k0 unit length l0 = � η/ξeff unit time t0 = η/ξeffD unit stress σ0 = ξeffD unit myosin motors concentration c0 = M √ η/ξeff unit density of cell-substrate bonds n0 = ξ/α TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Definitions of physical parameters and characteristic quantities in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Physical meaning Symbol unbinding rate ¯k0 = t0k0 coefficient for strain-rate-dependent unbinding ¯k1 = k1/t0 binding rate ¯kon = kont0/n0 base actin polymerization speed ¯v(0) p = v(0) p l0/t0 coefficient for stress-dependent actin polymerization ¯v(1) p = v(1) p l0 coefficient for cell polarization effect on actin polymerization ¯v(2) p = v(2) p t0/l0 contractility ˜χ = c0χ/σ0 cell elastic constant K = γl0/σ0 drag coefficient ˜ξ = ξ/(ξ + αkon/k0) TABLE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Definitions of dimensionless parameters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' its typical value ∼ 1/min [3], and we chose ¯k0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' ¯kon is chosen to be 6 for most simulations and the dimensionless density of bonds in the absence of mechanosensitivity is of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The dimensionless natural length of the cell L0 = 1 for a typical cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This means that in the absence of adhesion complexes, drag and viscous forces are of the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The dimensionless total number of myosin motors in the cell c0L0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3 S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' SIMULATION RESULTS The motility phase diagrams in the plane spanned by ˜χ and k1 are presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Here Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S1 shows the phase diagram in the plane spanned by v(0) p and k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' From this figure we can see how the actin polymerization rate affects the motility behavior of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is clear that the moving state of a cell with small k1 is that with a constant velocity, while the moving state of a cell with large k1 is periodic back-and-forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This is in agreement with the phase diagrams in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, there are several complex motility patterns between these two states, including stick-slip motion and behaviors that can be seen as combinations of back-and-forth and stick- slip movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The trajectories for stick-slip movement and periodic back-and-forth movement with stick-slip are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 1(c)(d) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The trajectories for zig-zag movement with stick-slip and double-period back-and-forth motion are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 4(a)(b) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' ��� ��� ��� ��� � ��� � ��� ��� ��� ��� ��� � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Phase diagram for the motility behavior predicted by the reduced model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, ˜χ = 14, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The cells show the following motility behaviors: constant velocity motion (green diamonds), periodic back-and-forth movement (blue circles), stick-slip movement (orange triangles), periodic back-and-forth movement with stick- slip (empty gray circles), zig-zag movement with stick-slip (empty purple diamonds), and double-period back-and-forth motion (brown squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Figure S2 shows the distribution of adhesion complexes and myosin motors for a cell that undergoes stick-slip movement and periodic back-and-forth movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Similar to the distributions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 2 of the main text, for a moving cell, myosin motors are always located relatively close to the trailing end, and the density of adhesion complexes is always higher in a region close to the leading end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S4 � �� �� � � � �� � ��� ��� ��� � (a) � �� �� � � � �� � � � � �� � � (b) ��� ��� ��� � � � �� � ��� ��� ��� � (c) ��� ��� ��� � � � �� � � � � �� � � (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Distribution of adhesion complexes and myosin motors for K = 100, ˜ξ = 1/3, kon = 6, k0 = 3, v(0) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='2, v(1) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='5, v(2) p = 2, and (a)(b) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='12, ˜χ = 18, (c)(d) k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='25, ˜χ = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S5 S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' THE SIMPLIFIED MODEL Since the adhesion complexes are concentrated close to the cell ends, to simplify the analysis, we assume nb(x, t) = Nfδ(x − xf) + Nbδ(x − xb), (S7) where xf = l+ − ǫ, xb = l− + ǫ, and ǫ is a very small length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Stress field and flow field Because there are no adhesion complexes in xb < x < xf, the dimensionless momentum equation in this region is ∂xσ = ˜ξv, σ = ∂xv + ˜χc, xb < x < xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S8) Integrating the full momentum equation from l+(−) to xf(b), we find σf = −K(l − l0) − ˜ξNfvf, σb = −K(l − l0) + ˜ξNbvb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S9) Here vf = v(xf, t) ≈ dl+/dt − v+ p , and vb = v(xb, t) ≈ dl−/dt + v− p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The solution of stress from these equations is [4] σ(x, t) = σf sinh � ˜ξ1/2(x − xb) � sinh � ˜ξ1/2(xf − xb) � + σb sinh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(xf − xb) � + ˜χ˜ξ1/2 � xf xb G(x, x′)c(x′, t)dx′, (S10) where G(x, x′) = sinh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(x′ − xb) � sinh � ˜ξ1/2(xf − xb) � − Θ(x′ − x) sinh � ˜ξ1/2(x′ − x) � , (S11) Θ(x) is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This leads to the following expression for the flow field, v(x, t) = 1 ˜ξ1/2 \uf8f1 \uf8f2 \uf8f3σf cosh � ˜ξ1/2(x − xb) � sinh � ˜ξ1/2(xf − xb) � − σb cosh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(xf − xb) � + ˜χ � xf xb ∂xG(x, x′)c(x′, t)dx′ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S12) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Myosin concentration Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S12) into the advection-diffusion for myosin concentration, one obtains the following equation which does not have an explicit dependence on the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' ∂tc(x, t) = D∂2 xc − 1 ˜ξ1/2 ∂x \uf8f1 \uf8f2 \uf8f3 \uf8ee \uf8f0σf cosh � ˜ξ1/2(x − xb) � sinh � ˜ξ1/2(xf − xb) � − σb cosh � ˜ξ1/2(xf − x) � sinh � ˜ξ1/2(xf − xb) � \uf8f9 \uf8fb c(x, t) +˜χ � xf xb c(x, t)∂xG(x, x′)c(x′, t)dx′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S13) S6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Velocity and length of the cell The velocity of the cell Vcell = 1 2 � dl+ dt + dl− dt � is Vcell = 1 2˜ξ1/2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � (σf − σb) + ˜χ 2 � xf xb sinh � ˜ξ1/2(xf − x′) � − sinh � ˜ξ1/2(x′ − xb) � sinh � ˜ξ1/2L � c(x′, t)dx′ +v+ p − v− p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S14) The evolution of the length of the cell dL dt = dl+ dt − dl− dt obeys dL dt = 1 ˜ξ1/2 cosh � ˜ξ1/2L � − 1 sinh � ˜ξ1/2L � (σf + σb) − ˜χ � xf xb sinh � ˜ξ1/2(xf − x′) � + sinh � ˜ξ1/2(x′ − xb) � sinh � ˜ξ1/2L � c(x′, t)dx′ +(v+ p + v− p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S15) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Symmetries of the system It is helpful to introduce the following variables N = Nf + Nb, ∆N = Nf − Nb, vp = v+ p + v− p 2 , ∆vp = v+ p − v− p , σS = σf + σb 2 = −K(L − L0) − ˜ξ 2 � N �dL/dt 2 − vp � + ∆N � Vcell − ∆vp 2 �� , σA = σf − σb 2 = − ˜ξ 2 � N � Vcell − ∆vp 2 � + ∆N �dL/dt 2 − vp �� , (S16) and y = x − l+ + l− 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S17) Notice that dL/dt, N, vp, and σS are symmetric under spatial inversion (y → −y), while Vcell, ∆N, ∆vp, and σA are antisymmetric under spatial inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The velocity of the cell Vcell can be expressed in terms of these parameters and variables that have clear parity signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, notice that only the part of c(y) that is anti-symmetric under y → −y contribute to the ˜χ-dependent term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S14), and for a slow-crawling cell this part should be significant only in the small |y| region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Thus by expanding the ˜χ-dependent term of Vcell to the leading order in y, one finds that Vcell = 1 2 \uf8ee \uf8f0∆vp + 2 ˜ξ1/2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � σA − ˜χ˜ξ1/2 cosh � 2˜ξ1/2L � sinh � ˜ξ1/2L � yc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' \uf8f9 \uf8fb , where yc ≡ � L/2 −L/2 y c(y, t) dy, (S18) and “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='..” represents terms of higher order in this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The expression for Vcell can be further simplified by taking ∆vp ≈ βVcell (β is independent of Vcell) for a slow crawling cell and substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S16) for σA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Finally, one S7 obtains Vcell = − ˜ξ1/2 1 − β/2 \uf8ee \uf8f01 + ˜ξ1/2 2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � N \uf8f9 \uf8fb −1 × \uf8ee \uf8f01 2 cosh � ˜ξ1/2L � + 1 sinh � ˜ξ1/2L � �1 2 dL dt − vp � ∆N + ˜χ cosh � ˜ξ1/2L/2 � sinh � ˜ξ1/2L � yc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S19) This expression tells us that Vcell is nonzero only when ∆N (asymmetry in the distribution of adhesion complexes) or yc (asymmetry in the distribution of myosin motors) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Similar calculation leads to the following expression for the evolution of the length of the cell, dL dt = 2vp + � 1 + ˜ξ1/2 2 cosh(˜ξ1/2L) − 1 sinh(˜ξ1/2L) N �−1 × \uf8f1 \uf8f2 \uf8f3 ˜ξ−1/2 cosh(˜ξ1/2L) − 1 sinh(˜ξ1/2L) � −2K(L − L0) + ∆N � 1 − β 2 � Vcell � − 2˜χ sinh � ˜ξ1/2L/2 � sinh � ˜ξ1/2L � Ctot \uf8fc \uf8fd \uf8fe , (S20) where Ctot ≡ � L/2 −L/2 c(y, t)dy ≡ 1 (S21) is the total amount of myosin motors in the cell, which is unity in our dimensionless expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Note that all terms on the right-hand side of dL/dt are even under y → −y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The simplified model and bifurcations From the previous analysis, it is clear that symmetry under y → −y plays an important role in Vcell and dL/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Based on these observations, a simplified model is proposed for slow-crawling cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, the evolution equations of Nf and Nb are dNf dt = kon − (k(0) off + k(1) off yc)Nf, dNb dt = kon − (k(0) off − k(1) off yc)Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This leads to dN dt = 2kon − k(0) off N − k(1) off yc ∆N, d∆N dt = −k(0) off ∆N − k(1) off Nyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S22) We expect k(1) off > 0 because a cell moving at constant velocity in the +x direction should have yc < 0 and ∆N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Next, the following evolution equation for yc is proposed dyc dt = −Γ � −(˜χ − ˜χc)yc − a∆N∆N + a3y3 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S23) The above equation describes a cell that becomes polarized (yc ̸= 0) when ˜χ is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, a∆N tells us how nonzero ∆N affects the evolution of yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In general, ˜χc, a∆N, and a3 all depend on L and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' a3 > 0 such that yc remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' To focus on the physics that are most relevant to the transitions between different motility behaviors, we neglect the N-dependencies in ˜χc and a∆N as they do not change the symmetry properties of the evolution equation of yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' This S8 approximation is expected to be suitable for slow-moving cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Furthermore, the L-dependencies of all coefficients in our simplified model can be neglected by considering the large-K regime such that L → L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' In this regime, Vcell = ˜ξ1/2 (1 − β 2 ) � 1 + ˜ξ1/2 2 cosh(˜ξ1/2L0)+1 sinh(˜ξ1/2L0) N � �� 1 2 cosh(˜ξ1/2L0) + 1 sinh(˜ξ1/2L0) vp � ∆N − � ˜χcosh(˜ξ1/2L0/2) sinh(˜ξ1/2L0) � yc � ≡ 1 1 − β/2(λν1vp∆N − ˜χλν2yc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S24) Here λν1vp and λν2 are N-dependent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Many interesting features of the system described by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S22)(S23) can be studied analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' First, the steady- state solutions include the rest-state solution ∆N = yc = 0, N = 2kon k(0) off ≡ N0, (S25) and solutions for a cell moving in the ± x-direction with a constant velocity yc = ∓ � p4 + p2 1p2 − � (p4 + p2 1p2)2 − 4p2 1p4(p2 − p1p3N0) 2p2 1p4 , ∆N = − p1N0 1 + (p1yc)2 yc, N = N0 1 − (p1yc)2 , (S26) where p1 = k(1) off /k(0) off , p2 = Γ(˜χ− ˜χc)/k(0) off , p3 = Γa∆N/k(0) off , and p4 = Γa3/k(0) off .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Further checking the linear stability of the rest state shows that the transition from the rest state to the state with constant velocity is a pitchfork bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' On the other hand, the transition from the rest state to the periodic back-and-forth movement is a Hopf bifurcation: Pitchfork bifurcation (rest/constant-velocity transition) happens when ˜χ = ˜χc + 2a∆N konk(1) off (k(0) off )2 (S27) and Γ(˜χ − ˜χc) − k(0) off < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S28) Hopf bifurcation (rest/back-and-forth-motion transition) occurs when ˜χ = ˜χc + k(0) off /Γ (S29) and ˜χ − � ˜χc + 2a∆N konk(1) off (k(0) off )2 � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' (S30) The phase diagram for the motility behavior predicted by this phenomenological model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' It is qualitatively similar to the phase diagrams of the active gel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The differences are likely due to the approximations we made when constructing the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' For example, assuming a constant cell length and assuming that the dynamics of yc are independent of N and L are likely to have some effects on the detailed shape of the phase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S9 � � � � � � � �� � � � � � � ��� ��� ��� ��� ��� ��� ��� ��� ��� � ��� ��� �� ��� ��� ���� ���������� �������������� ���������� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Phase diagram for the motility behavior predicted by the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The following motility patterns are found: a cell at rest (red squares), a cell moving at constant velocity (green diamonds), a cell performs stick-slip movement (orange triangles), a cell performs back-and-forth movement with stick-slip (at k(1) off /k(0) off slightly greater than those orange triangles so that we cannot show), and a cell performs periodic back-and-forth movement (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The boundary between the rest and constant velocity movement is ˜χ = ˜χc + 2a∆N konk(1) off � k(0) off �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' The boundary between the rest and periodic back-and-forth movement states is ˜χ = ˜χc + k(0) off /Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Luo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Mohan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Srivastava, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Ren, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Iglesias, and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [3] Y-L Wang, Reorganization of actin filament bundles in living fibroblasts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Cell Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=', 99, 1478 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Recho and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' Truskinovsky, “Cell locomotion in one dimension,” in Physical Models of Cell Motility, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'} +page_content=' 135–197, Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQff_zo/content/2301.01463v1.pdf'}