Mutation at Expanding Front of Self-Replicating Colloidal Clusters

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Mutation at Expanding Front of Self-Replicating Colloidal Clusters

Hidenori Tanaka, Zorana Zeravcic, and Michael P. Brenner

Phys. Rev. Lett. 117, 238004 – Published 2 December 2016

Abstract

We construct a scheme for self-replicating square clusters of particles in two spatial dimensions, and validate it with computer simulations in a finite-temperature heat bath. We find that the self-replication reactions propagate through the bath in the form of Fisher waves. Our model reflects existing colloidal systems, but is simple enough to allow simulation of many generations and thereby the first study of evolutionary dynamics in an artificial system. By introducing spatially localized mutations in the replication rules, we show that the mutated cluster population can survive and spread with the expanding front in circular sectors of the colony.

Received 19 July 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.238004

Physics Subject Headings (PhySH)

Biomimetic & bio-inspired materials Evolutionary & population dynamics Mutation Replication Self-assembly

Colloids

Molecular dynamics

Interdisciplinary PhysicsCondensed Matter, Materials & Applied PhysicsPhysics of Living Systems

Authors & Affiliations

Hidenori Tanaka^1,2,*, Zorana Zeravcic^1,3, and Michael P. Brenner^1,2

¹Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
²Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA
³Soft matter and chemistry laboratory, ESPCI PSL Research University, 75005 Paris, France

^*tanaka@g.harvard.edu

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Vol. 117, Iss. 23 — 2 December 2016

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Images

Figure 1
(a) Self-replication scheme of square clusters with $n_{c} = 3$ . The reaction starts with two square clusters composed of $A$ and $B$ particle species, having permanent bonds (stage I). Complementary monomers ( $A^{'}$ , $B^{'}$ ) attach to the clusters (stage II) and attraction among the attached monomers leads to templating of another 4-mer structure (stage III). The thus formed complementary cluster templates a new cluster in the same way, thereby closing the hypercycle (stages IV to VI). (b) Interaction matrix of particle species. Blue matrix entries represent attractive interaction, and white entries represent repulsive force, while the inscribed number specifies the species valence.
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Figure 2
Brownian dynamics simulation of self-replicating colloidal clusters using 1652 colloidal particles. (a) A snapshot from our simulation: Particles are colored according to their species [Fig. 1] if they comprise a cluster; otherwise they are gray. This snapshot shows successful replication of the desired 4-mer (square) cluster as well as other undesired 5-mer and 6-mer structures. (b) Example of frequent replication pathways leading to formation of each of the 4-, 5-, and 6-mers. (c) Population of 4-mers, 5-mers, and 6-mers as a function of time.
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Figure 3
Simulation snapshots taken at $Δ t = 100 d^{2} / D$ intervals. Clusters with at least two attached monomers are colored red (active parents) and clusters with less than two attached monomers are colored blue (inactive parents). Dashed lines ( $\propto t$ ) are guides to the eye.
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Figure 4
(a) Log-log plot of the total population as a function of time. The solid line is the average over 10 independent simulations. The shaded region shows 1 standard deviation above and below the average, while the dashed line ( $\propto t^{2}$ ) is the best linear fit. (b) Radius of the circular colony as a function of time. The solid line is the average over 10 independent simulations. The shaded region shows 1 standard deviation above and below the average. The dashed line is the best linear fit, suggesting emergence of the expanding front with constant velocity.
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Figure 5
(a) The second snapshot ( $t = t_{0} + Δ t$ ) of Fig. 3 is shown, where one square cluster (orange) at the boundary is mutated by assigning it $n_{c} = 4$ . All other clusters (green) retain $n_{c} = 3$ . (b1)–(b2) and (c1)–(c2) are results from two independent simulations started from (a). The mutated progeny either dominates in a circular sector of the colony (b1) and survives with expanding front (b2), or stays within the colony bulk (c1) and stops reproducing (c2).
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