Abstract
A generically observed mechanism that drives the self-organization of living systems is interaction via chemical signals among the individual elements -- which may represent cells, bacteria, or even enzymes. Here we propose a novel mechanism for such interactions, in the context of chemotaxis, which originates from the polarity of the particles and which generalizes the well-known Keller--Segel interaction term. We study the resulting large-scale dynamical properties of a system of such chemotactic particles using the exact stochastic formulation of Dean and Kawasaki along with dynamical renormalization group analysis of the critical state of the system. At this critical point, an emergent "Galilean" symmetry is identified, which allows us to obtain the dynamical scaling exponents exactly; these exponents reveal superdiffusive density fluctuations and non-Poissonian number fluctuations. We expect our results to shed light on how molecular regulation of chemotactic circuits can determine large-scale behavior of cell colonies and tissues.
Highlights
Characterizing the emergence of macroscopic properties in colonies of prokaryotic [1,2] and eukaryotic [3] cells based on the complicated chemical interactions among the individuals in the colony is a long-standing endeavor in various areas of biology such as morphogenesis [4,5,6], tissue growth and homeostasis [7], wound healing [8], and cancer metastasis [9,10]
This section addresses the outcome of the renormalization group (RG) analysis of the Langevin equation (19) by, first, describing the RG flow diagrams that are obtained within our one-loop computation and, discussing the exact exponents that characterize the scaling laws of the critical system
By assuming that the density fluctuations ρ are sufficiently small compared to the average density C0 (i.e., ρ C0), one discards the symmetry-breaking term ∇ · (μ3ρ∇(∇φ)2) from the expansion (which would appear in the Langevin dynamics (19) if the full particle current (18) is kept) since this term scales as ρ3 and its coupling μ3 is irrelevant under RG
Summary
Characterizing the emergence of macroscopic properties in colonies of prokaryotic [1,2] and eukaryotic [3] cells based on the complicated chemical interactions among the individuals in the colony is a long-standing endeavor in various areas of biology such as morphogenesis [4,5,6], tissue growth and homeostasis [7], wound healing [8], and cancer metastasis [9,10]. The critical dynamics is analyzed using a dynamical renormalization group (RG) treatment [64,65,66] to obtain the emergent macroscopic properties of the chemotactic system based on the interactions between its individuals In these stochastic field equations, we identify an emergent symmetry, which turns out to coincide with the “Galilean” symmetry known in the apparently unrelated context of the Kardar-Parisi-Zhang (KPZ) equation [65,67,68]. The analysis of the scaling properties of the stochastic evolution equation is performed both with a nonconserved noise, relevant in the case where the number of particles is conserved only on average, and with a conserved noise In both cases, the exact exponents we obtain predict superdiffusion at the critical state, while the magnitude of the fluctuations of the particle number depends on the nature of the noise: A conserved noise suppresses these fluctuations and the distribution becomes hyperuniform, whereas a nonconserved noise enhances the fluctuations and leads to giant number fluctuations. II (Appendix B), the validity of detailed balance in our chemotactic field theory (Appendix C), the gradient expansion and power counting (Appendix D), the details of the RG calculations (Appendices E and F), the analysis of the RG flows in various spatial dimensions (Appendix G), and the thorough discussion of the moment expansion, anticipated in Appendix B, for a more general chemotactic model including self-propulsion and nematic alignment of the particles (Appendix H)
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