Collective Signal Processing in Cluster Chemotaxis: Roles of Adaptation, Amplification, and Co-attraction in Collective Guidance
[['Many individual cells , including white blood cells and bacteria , chemotax—sensing and following gradients of signals .', 'Some cells , though , are not loners—they migrate collectively—and cells traveling in clusters and sheets during development must chemotax together .', 'Many experiments [1–5] have shown that clusters can have capabilities that single cells lack: in particular , clusters of cells can follow a gradient even when single cells do not .', 'How can cells work together to follow a gradient that each individual cell is incapable of sensing ?', 'How can cells integrate data from across the cluster to improve their gradient sensing abilities ?', 'Is cluster chemotaxis essentially different than single-cell chemotaxis ?', 'The simplest possibility , that cells just spatially average the gradient signal acting independently on each of them and thereby achieve a more accurate sensing capability , is ruled out , at least for lymphocytes , by experiments that show clusters can travel in the direction opposite to that of single cells [1] .', 'A different possible explanation relies on the qualitative idea of collective guidance [6] , in which a cluster of cells can gain a direction even though each of its individual cells senses only the level of signal , and not its gradient .', 'To make this notion more quantitative , we have recently introduced such a model of collective guidance in the context of neural crest cells where the cluster’s directionality comes from a regulation of contact inhibition of locomotion ( CIL ) [7]; a related model was also proposed for clusters of lymphocytes [1] and extended for studying border cell migration [8] .', 'However , our current understanding of collective guidance and how collective chemotaxis occurs without single-cell gradient sensing does not account for the possibility of response coordinated by chemical signaling between cells .', 'Our minimal model of collective guidance posits that each cell reacts only to the local chemoattractant and the physical presence of its neighbors [7] .', 'More complicated signal processing could take place on the cluster scale if cells use signaling molecules to communicate with each other to collectively process the information contained in the chemoattractant gradient , as was recently suggested to be the case in branching morphogenesis [4 , 9] .', 'It is therefore important to ask: What experimental signatures would tell us if this were happening , and how would this signal processing change the efficiency of the cluster’s movement ?', 'Can collective signal processing overcome shallow gradients seen in vivo ( e . g . [10] ) , amplifying differences in cluster behavior between the front and the back ?', 'In minimal models of collective guidance [1 , 7] , the cluster moves by a tug of war , and is likely under tension .', 'Nevertheless , collective chemotaxis can also occur in the absence of strong adhesion [2] .', 'How does this happen ?', 'We will address all of these questions in this paper .', 'Our initial focus will be on understanding in vitro experiments in relatively controlled environments [1 , 2] , especially experiments on explants of neural crest cells , and using these results to develop a useful quantitative framework for the study of collective guidance more generally , including collective chemotaxis in vivo [11 , 12] .', 'To understand why clusters of explanted neural crest cells chemotax where single cells do not [2] , we will analyze both short-range interactions between cells and long-range interactions mediated by chemical secretions .', 'The primary short-range interaction between neural crest cells are cadherin-mediated adhesion and contact inhibition of locomotion ( CIL ) .', 'CIL results in cells repolarizing away from each other after contact .', 'CIL in tissues may be regulated by the type of cadherin expressed , as well as being linked to mechanical force between cells [13–16] .', 'Many possible molecular mediators of CIL have been established , including the non-canonical Wnt-planar cell polarity pathway and ephrin signaling [17 , 18] .', 'Within this paper , we will take a phenomenological approach to modeling CIL , describing its consequences rather than its molecular origin .', 'We first study models of biochemical processing of the chemoattractant signal within the cell cluster , assuming strong cell-cell adhesions as in our earlier model [7] .', 'We treat the possibility of gradient sensing via cell-cell communication , using a mechanism that allows adaptation , i . e . the cluster’s response becomes insensitive to the overall level of the signal S ( r ) .', 'We do this using a local excitation , global inhibition ( LEGI ) scheme [19] .', 'This model is supported by recent experiments on collective gradient sensing in branching morphogenesis , which identify gap-junction-mediated communication between cells as a critical aspect of collective gradient sensing [4 , 9] .', 'We also consider the possibility of cluster-level amplification of a sensed gradient , where relatively small changes in the chemoattractant signal S ( r ) across the cluster are amplified into much larger changes in the response level .', 'With both adaptation and a switch-like amplification , we find that clusters of an optimal size are more efficient at chemotaxing than either smaller or larger clusters .', 'Amplification of the external signal allows clusters to develop a large velocity even in a shallow gradient .', 'We argue , based on simple scaling principles , that sufficiently large clusters with only short-range adhesion undergoing collective guidance would be expected to either fragment or become increasingly slow .', 'We then show that if the cohesion of a cluster is not controlled by local cell-cell adhesion , but rather by chemotaxis toward a secreted signal ( or “co-attraction” [20] ) , a cluster of cells can undergo collective guidance by regulation of CIL even if cells are not in continuous contact .', 'We show how co-attraction and regulated CIL interact in order to create robust chemotaxis .', 'In the presence of co-attraction , new behaviors , including persistent cluster rotation , may emerge .', 'We provide an extensive characterization of the transition to rotation , and how rotation can alter the efficiency of gradient-sensing clusters .'], ['We want to model the collective guidance of a cluster of cells exposed to a chemical gradient S ( r ) .', 'We use the experiments of [2] on neural crest explants responding to Sdf1 gradients as a guide to determine the features we include as well as the model parameters , though we expect our results to be more generally applicable as well .', 'There are four major elements of a model of this process: 1 ) single-cell dynamics , 2 ) physical interactions between cells and contact-range effects like contact inhibition of locomotion , 3 ) the response of the cells to the chemical S ( r ) , and 4 ) chemical communication and signaling between cells .', 'We use a two-dimensional stochastic particle model to describe cells exposed to a chemical gradient S ( r ) .', 'We describe each cell i with a position ri and a polarity pi .', 'The cell polarity indicates the cell’s direction and propulsion strength ,', 'i . e .', 'the velocity with which it would travel in the absence of additional forces; we thus define pi so that an isolated cell with polarity pi has velocity pi .', 'The cell’s motion is overdamped , so physical forces like cell-cell adhesion and exclusion change the cell’s velocity—the velocity of the cell is pi plus the net force the other cells exert on it , ∑ j ≠ i F i', 'j . We model chemically-induced effects like CIL as altering a cell’s biochemical polarity pi .', 'Our model is then:', '∂ t r i = p i + ∑ j ≠ i F i j ( 1 ) ∂ t p i = - 1 τ p i + σ ξ i ( t ) + β i ∑ j ∼ i r ^ i j + χ ∇ c ( r i ) | ∇ c | Θ ( | ∇ c | - g 0 ) ( 2 )', 'where Fij are intercellular forces , e . g . cell-cell adhesion and volume exclusion , and ξi ( t ) are fluctuating , temporally uncorrelated noise terms that are Gaussian with 〈 ξ μ i ( t ) ξ ν j ( t ′ ) 〉 = 2 δ μ ν δ i j δ ( t - t ′ ) , where the Greek indices μ , ν run over the dimensions x , y .', 'The first two terms on the right of Eq 2 are a standard Ornstein-Uhlenbeck model [28 , 29]: pi returns to zero with a timescale τ , but is pushed away from zero by the fluctuating noise ξ ( t ) .', 'This models a cell that has a motion that is only persistent over a time of τ .', 'Throughout this paper , we choose our units to be defined by the typical parameters of neural crest cells .', 'With this in mind , we take our length scale to be the typical equilibrium cell-cell separation and our time scale to be the relaxation time—this corresponds to setting the cell diameter to be unity and the relaxation time τ = 1 .', 'To convert between these simulation units and real units , we use values estimated from the experiments of [2]: typical equilibrium cell-cell separation is 20 μm and the typical time over which a cell reorients is roughly 20 minutes ,', 'i . e .', 'τ = 20 minutes in real units .', 'Within the simulation units we have chosen , measured neural crest cell velocities are on the order of 1 , so we choose σ = 1 .', 'This choice means that the root mean square speed of an isolated cell is 〈 | V | 2 〉 1 / 2 = 2 1 / 2 σ τ 1 / 2 ≈ 1 .', '4 microns/minute , in good agreement with , e . g . [2] .', 'When we include adaptation , we assume that the kinetics of Eqs 7 and 9 are fast compared with the dynamics of interest , and set them to their steady states , assuming k−R ≫ kR and thus Ri = Ai , ss/Ii ( t ) .', 'We set the diffusion rate kD = 4 in our units , corresponding to a time for equilibration of a few minutes , consistent with experiments using FRAP to see equilibration of fluorescent dyes across gap junctions [39] .', 'However , we note that this rate can depend on the identity of the inhibitor , and may also be regulated [40 , 41] .', 'We set the rates of generation and decay of the inhibitor to be kI = k−I = 1; this is discussed more in the adaptation section .', 'A complete list of parameters and their justifications is included in the Supplementary Information , Table S1 .', 'We integrate Eqs 1 , 2 and 7–9 explicitly with an Euler-Maruyama integrator [42] .', 'The time step varies: for rigid clusters with high adhesion , we choose Δt = 1 × 10−4 , and for co-attraction simulations we choose Δt = 1 × 10−3 .', 'Further details about time step selection as well as source code are available in the Supplementary Information .'], ['In our earlier paper [7] , we studied a minimal version of the model described above , with no co-attraction ( χ = 0 ) and no adaptation or amplification , i . e . β i = β ¯ S ( r i ) .', 'We briefly note a few results from that paper here , as in some limits , our more complex model will reduce to this model .', 'Under assumptions of cluster rigidity and slow reorientation , the mean drift of a cluster of cells obeying Eqs 1 and 2 is given by', '〈 V 〉 c ≈ β ¯ τ M · ∇ S ( 10 )', 'with the approximation true for S ( r ) ≈ S0 + r · ∇S .', '〈⋯〉c is an average over the fluctuating pi but with fixed configuration and orientation of cells ri .', 'The matrix M depends only on the configuration of cells; formulas for many cluster shapes and sizes are given in [7] .', 'Mean cluster velocity 〈Vx〉 saturates at large number of cells N . This arises because we have the difference in signal between the front and the back growing as the cluster radius ( ∼ N ) , while the perimeter of the cluster also grows as N . The force on the cluster then grows as N at large N , while the effective friction of the cluster grows independently with the number of cells , as N—hence the net velocity should behave as ∼N1/2 × N1/2/N ∼ 1 at large N . ( Similar scaling arguments are found for the circular cluster limit in [1] . ) As we move beyond the minimal model , these scaling assumptions may break down , and therefore larger clusters will not necessarily have saturating velocities .', 'Ref .', '[7] also provides analytic results for the chemotactic index CI– a measure of the directionality of the cluster motion .', 'This is commonly defined as the ratio of the distance traveled in the direction of the gradient ( the x direction ) to the total distance traveled .', 'To clarify how we average over many realizations of a path , we define CI = 〈Vx〉 / 〈|V|〉 .', 'Until this point , we have only looked at highly adherent , effectively rigid clusters .', 'However , collective cell migration can also occur with a high degree of fluidity and cell-cell rearrangement [12 , 53–60] .', 'In addition , we have until now assumed that the only attraction between cells is short-range , representing cell-cell adhesion .', 'However , neural crest cells also attract one another through chemical secretions , which can control the extent of cluster directionality and cohesion [20 , 27]—and many other cell types also chemotax toward secretions [61 , 62] .', 'We extend our model to allow for this possibility , and show that clusters of cells that cohere via co-attraction can also be directed by collective guidance .', 'These clusters need not be rigid , and can have significant re-arrangement or even only transient contacts .', 'In this section , we will treat clusters with co-attraction ( χ ≠ 0 ) , but assume only the minimal model of signal processing , with the CIL susceptibility β i = β ¯ S ( r i ) .'], ['Other variants of stochastic particle models have been used to model collective cell migration , ranging from models that use single particles to represent cells [30 , 53 , 63 , 84–86] to those that use more detailed representations of cells with either multiple particles or additional details of cell shape [87–91] .', 'Other techniques , such as the Cellular Potts Model [58 , 75 , 92] and phase field models [23 , 93 , 94] have also been developed to study collective cell migration with significantly greater levels of detail on the cell’s shape and its internal biochemistry .', 'Because emergent collective guidance has had only limited quantitative models in the past [1 , 7] , we have chosen our cell models to be as minimal as possible , in an attempt to focus on the essential aspects of collective guidance .', 'Earlier models have been created to study neural crest chemotaxis in vivo [81 , 83 , 95]; however , these have explicitly described chemotaxis arising from a “follow-the-leader” mechanism where single leader cells can sense a gradient [96] , rather than through the collective mechanism we study here , where individual cells need not sense the gradient level .', 'We also mention that unlike many of the models discussed above , our model does not include an interaction designed to align a cell’s polarity with its neighbors’ motion [53 , 63 , 75] or its own velocity or displacement [30 , 58 , 87 , 88] , and these mechanisms are not necessary for the effects we describe here .', 'Competition between the collective guidance mechanism and alignment mechanisms may be an interesting area for future study .', 'Our stochastic interacting particle model is relatively simple , which allows us to in some cases derive analytic results [7] .', 'Many extensions of this approach are possible .', 'Our model could be developed further for more quantitative comparisons by careful measurement of single-cell statistics in or out of a chemoattractant gradient [28 , 97]; this could lead to nonlinear or anisotropic terms in Eq 2 .', 'Our description of contact inhibition of locomotion has also assumed , for simplicity , that contact with both the front and back of the cell is inhibitory; other possibilities may alter the collective dynamics of the cell cluster [23] .', 'Our main findings are:', '1 ) Cluster velocity and chemotactic index may reflect internal signal processing , and provide an experimental window into these processes .', '2 ) We expect sufficiently large clusters undergoing collective guidance to either become increasingly slow or break up .', '3 ) Strong adhesion between cells is not necessary for collective guidance to function if cells chemotax to a secreted molecule .', '4 ) A balance of this co-attraction and graded contact inhibition of locomotion are necessary for efficient chemotaxis .', '5 ) Co-attraction may also induce cluster rotation , and we have explicitly characterized the transition to rotation .', '6 ) The combination of cluster rotation and cluster chemotaxis may induce systematic drifts that depend on cluster rotation .']]
['Introduction', 'Model', 'Results', 'Discussion']
['Single eukaryotic cells commonly sense and follow chemical gradients , performing chemotaxis .', 'Recent experiments and theories , however , show that even when single cells do not chemotax , clusters of cells may , if their interactions are regulated by the chemoattractant .', 'We study this general mechanism of “collective guidance” computationally with models that integrate stochastic dynamics for individual cells with biochemical reactions within the cells , and diffusion of chemical signals between the cells .', 'We show that if clusters of cells use the well-known local excitation , global inhibition ( LEGI ) mechanism to sense chemoattractant gradients , the speed of the cell cluster becomes non-monotonic in the cluster’s size—clusters either larger or smaller than an optimal size will have lower speed .', 'We argue that the cell cluster speed is a crucial readout of how the cluster processes chemotactic signals; both amplification and adaptation will alter the behavior of cluster speed as a function of size .', 'We also show that , contrary to the assumptions of earlier theories , collective guidance does not require persistent cell-cell contacts and strong short range adhesion .', 'If cell-cell adhesion is absent , and the cluster cohesion is instead provided by a co-attraction mechanism , e . g . chemotaxis toward a secreted molecule , collective guidance may still function .', 'However , new behaviors , such as cluster rotation , may also appear in this case .', 'Co-attraction and adaptation allow for collective guidance that is robust to varying chemoattractant concentrations while not requiring strong cell-cell adhesion .']
['To get from one part of the body to another , single cells often follow chemical signals .', 'Sometimes , though , isolated cells ignore these signals , but a group of cells still manages to travel in a directed way .', 'How can this happen ?', 'We argue that if the signal changes how the cells interact with their neighbors , a cluster of cells can detect signals single cells ignore .', 'We use computational models to study how this can happen , and show that the speed of the cluster will depend on how the cells process the signal , as well as whether or not the cells are tightly connected to one another .', 'We also show if the cells are only loosely connected , and are attracted to a secreted molecule , cell clusters may develop rotation and other effects that will change how effectively they can sense signals .']
['velocity', 'cell physiology', 'cell motility', 'medicine and health sciences', 'classical mechanics', 'engineering and technology', 'nervous system', 'signal processing', 'junctional complexes', 'electrophysiology', 'neuroscience', 'cell polarity', 'gap junctions', 'stem cells', 'phase diagrams', 'developmental neuroscience', 'computer and information sciences', 'animal cells', 'neural crest', 'chemotaxis', 'physics', 'signal transduction', 'cellular neuroscience', 'neural stem cells', 'cell biology', 'anatomy', 'synapses', 'data visualization', 'physiology', 'biology and life sciences', 'cellular types', 'physical sciences', 'cell signaling', 'neurophysiology', 'motion']