Collective gradient sensing with limited positional information

Emiliano Perez Ipiña and Brian A. Camley

Phys. Rev. E 105, 044410 – Published 26 April 2022

Abstract

Eukaryotic cells sense chemical gradients to decide where and when to move. Clusters of cells can sense gradients more accurately than individual cells by integrating measurements of the concentration made across the cluster. Is this gradient-sensing accuracy impeded when cells have limited knowledge of their position within the cluster, i.e., limited positional information? We apply maximum likelihood estimation to study gradient-sensing accuracy of a cluster of cells with finite positional information. If cells must estimate their location within the cluster, this lowers the accuracy of collective gradient sensing. We compare our results with a tug-of-war model where cells respond to the gradient by polarizing away from their neighbors without relying on their positional information. As the cell positional uncertainty increases, there is a trade-off where the tug-of-war model responds more accurately to the chemical gradient. However, for sufficiently large cell clusters or sufficiently shallow chemical gradients, the tug-of-war model will always be suboptimal to one that integrates information from all cells, even if positional uncertainty is high.

Received 5 November 2021
Accepted 21 March 2022

DOI:https://doi.org/10.1103/PhysRevE.105.044410

Physics Subject Headings (PhySH)

Cell migration Chemotaxis Collective behavior

Physics of Living Systems

Authors & Affiliations

Emiliano Perez Ipiña ^*

Department of Physics & Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA

Brian A. Camley ^†

Department of Physics & Astronomy and Biophysics, Johns Hopkins University, Baltimore, Maryland 21218, USA

^*emperipi@jhu.edu
^†bcamley@jhu.edu

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Vol. 105, Iss. 4 — April 2022

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Images

Figure 1
(a) Cluster under a linear gradient and a representation of the two sources of uncertainties: sensing concentration and positional errors. (b) Gradient-sensing error increases with positional uncertainty. (c, d) Positional uncertainties becomes more dominant in steeper gradients. (c) Gradient-sensing error normalized by its value in the absence of positional errors, $σ_{g}^{2} (Δ r = 0)$ , as a function of the gradient steepness (blue solid line). The black dashed line indicates the limit where the gradient-sensing error is dominated by positional errors, and it follows $\propto g^{2} Δ r^{2}$ . (d) Gradient-sensing relative error $σ_{g} / g$ as a function of the gradient steepness. Parameters: number of cells $N = 37$ (corresponding to an hexagonal cluster of $Q = 3$ layers), $R_{cell} = 10 μ m$ , (b) $g = 0.005 μ m^{- 1}$ , (c, d) $Δ r = 10 μ m$ .
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Figure 2
Gradient-sensing errors for a cluster with a fraction $f$ of informed cells. (a) Informed cells are distributed over the cluster in three distinct manners: closer to the cluster's center, randomly, or closer to the cluster's edge. Underneath each example of cluster distribution we show two of the multiple equivalent ways of distributing the informed cells. (b, c) Gradient-sensing variance, $σ_{g}^{2}$ , normalized by the case of noninformed cells, $σ_{g_{0}}^{2} = σ_{g}^{2} (f = 0)$ , as a function of the fraction of informed cells (b) and the ratio of the two types of cell positional uncertainties (c). (d) Relative difference between informed cluster and the equivalent constant-positional-error cluster, $δ σ_{g} = \frac{σ_{g} (\bar{Δ r}) - σ_{g} (Δ r_{\inf}, Δ r)}{σ_{g} (Δ r_{\inf}, Δ r)}$ , as a function of the gradient steepness. $δ σ_{g} > 0$ ( $δ σ_{g} < 0$ ) indicates a lower (higher) gradient-sensing error for the cluster with the informed cells compared to the uniform positional error cluster. The dashed black line indicates $δ σ_{g} = 0$ . For panels (b)–(d) color codes are the same. Solid lines with symbols represent the three different distribution patterns: center (red squares), random (light blue crosses), and edge (dark blue circles). The dashed black line represents the equivalent one type of cells cluster with positional uncertainty equal to $\bar{Δ r} = f Δ r_{\inf} + (1 - f) Δ r$ . Parameters: cluster size $N = 91$ ( $Q = 5$ layers in the hexagonal cluster), $Δ r = 2 R_{cell}$ , (b) $Δ r_{\inf} = 0.5 Δ r$ , (c) $f = 0.25$ , (d) $f = 0.25, Δ r_{\inf} = 0.2 Δ r$ , and $\bar{Δ r} = 20 μ m$ . Curves show an average of 100 realizations.
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Figure 3
(a) Schematic cell cluster with a group of four informed cells located at a distance $r$ from the cluster's center and direction $θ$ with respect to the chemoattractant gradient. The small clusters on the right represent different small cluster configurations. (b) Relative gradient-sensing steepness (left panel) and direction, $δ σ_{ϕ} = \frac{σ_{ϕ} (\bar{Δ r}) - σ_{ϕ} (Δ r_{\inf}, Δ r)}{σ_{ϕ} (Δ r_{\inf}, Δ r)}$ , (right panel) errors when the informed group of cells is displaced in different directions: $θ = 0$ (red line and squares), $θ = π / 4$ (light blue line and crosses), and $θ = π / 2$ (dark blue line and circles). Same as in Fig. 2, $δ σ_{g, ϕ} > 0$ ( $δ σ_{g, ϕ} < 0$ ) represents a lower (higher) error relative to the constant-positional-error equivalent cluster. The dashed black line indicates $δ σ_{g, ϕ} = 0$ and also represents the equivalent cluster without informed cells and positional uncertainty $\bar{Δ r}$ . Parameters: cluster size $N = 91$ and $Δ r_{\inf} = 0.5 Δ r$ . Curves show an average of 100 realizations.
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Figure 4
Tug-of-war model. (a) Scheme representation of the tug-of-war model. (b) Gradient sensing variance as a function of the positional uncertainty for the tug-of-war model (red solid line) and the MLE model (blue solid line). The black dashed line marks the crossover and the trade-off positional uncertainty between the two models. (c, d) Trade-off value as a function of (c) the cluster number of cells and (d) the gradient steepness. (e) Phase diagram of the trade-off values for different clusters' sizes and gradient steepness. The dashed red line indicates a boundary given by $Δ r_{t} = 2 R_{cell}$ . Positional errors, $Δ r$ , and trade-off values, $Δ r_{t}$ , are rescaled in of cell's radius. Parameters: (b) same as Fig. 1, (c) $g = 0.005 μ m^{- 1}$ , and (d) $N = 37$ ( $Q = 3$ ).
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