Collective Dynamics of Dividing Chemotactic Cells

The large scale behavior of a population of cells that grow and interact through the concentration field of the chemicals they secrete is studied using dynamical renormalization group methods. The combination of the effective long-range chemotactic interaction and lack of number conservation leads to a rich variety of phase behavior in the system, which includes a sharp transition from a phase that has moderate (or controlled) growth and regulated chemical interactions to a phase with strong (or uncontrolled) growth and no chemical interactions. The transition point has nontrivial critical exponents. Our results might help shed light on the interplay between chemical signaling and growth in tissues and colonies, and in particular on the challenging problem of cancer metastasis.

Figure 1

Schematics of the model, showing (a) the interaction between the cells via a long-range field of emitted chemicals, and (b) the cell division (and death) process.

Figure 2

Properties of the fixed point and the accompanying dynamical phase transition. (a) The flow around the fixed point in (${\nu }_1$, ${\nu }_2$) space. (b) The two parameter regions for ${\nu }_1$, ${\nu }_2$ for different dimensions. Above the separatrix (dashed line), a perturbatively not accessible fixed point will control the flow, whereas below it the flow will converge to the stable fixed point. The solid line represents the critical point that corresponds to the phase transition, and can be experimentally approached by tuning any of the parameters involved, e.g., the growth rate or the rate of release of the chemicals.

Figure 3

One-loop diagrams (a) for the response function $G(\widehat{k})$, (b) the noise correlator $\mathcal{D}(k)$, and (c) the vertex function, in terms of the bare quantities defined in (d), namely, $G_0(\widehat{k})=[i\omega +D{k}^2+\theta {]}^{-1}$ and ${\mathcal{D}}_0(k)=D_0+D_2{k}^2$, where $\widehat{k}≔(\mathbf{k},\omega )$.

Figure 4

(a) The fixed point is stable in the space of equilibrium and nonequilibrium noises for $d\le 4$. (b) The values for the exponents $z$ and $\chi$ corresponding to the stable fixed point. For $d=(1,2,3)$ dimensions, the values are reported in Table 1.