Self-replication with magnetic dipolar colloids

Joshua M. Dempster, Rui Zhang, and Monica Olvera de la Cruz

Phys. Rev. E 92, 042305 – Published 8 October 2015

Abstract

Colloidal self-replication represents an exciting research frontier in soft matter physics. Currently, all reported self-replication schemes involve coating colloidal particles with stimuli-responsive molecules to allow switchable interactions. In this paper, we introduce a scheme using ferromagnetic dipolar colloids and preprogrammed external magnetic fields to create an autonomous self-replication system. Interparticle dipole-dipole forces and periodically varying weak-strong magnetic fields cooperate to drive colloid monomers from the solute onto templates, bind them into replicas, and dissolve template complexes. We present three general design principles for autonomous linear replicators, derived from a focused study of a minimalist sphere-dimer magnetic system in which single binding sites allow formation of dimeric templates. We show via statistical models and computer simulations that our system exhibits nonlinear growth of templates and produces nearly exponential growth (low error rate) upon adding an optimized competing electrostatic potential. We devise experimental strategies for constructing the required magnetic colloids based on documented laboratory techniques. We also present qualitative ideas about building more complex self-replicating structures utilizing magnetic colloids.

Received 10 March 2015
Revised 10 June 2015

DOI:https://doi.org/10.1103/PhysRevE.92.042305

Authors & Affiliations

Joshua M. Dempster¹, Rui Zhang², and Monica Olvera de la Cruz^1,2,3

¹Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
²Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
³Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA

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Vol. 92, Iss. 4 — October 2015

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Figure 1
The simple dipole replication scheme. (a) Detailed sketch of the dipole model used for self-replication. Binding sites (dots on the side) interact via an attractive infinite potential of range $2 r_{b}$ with the minimum at $r_{b}$ . Black arrows indicate magnetic dipole moment direction $\hat{μ}$ . (b) A dimer template which will be replicated. If the external field strength $H$ is weak compared to $k$ , then the two magnetic moments in the dimer antialign. (c) If no colloids are bound, then the preferred configuration for low dipole concentrations in a weak field is a line. Binding sites are never close enough to form a new dimer. (d) The four-step replication cycle. In steps (i)–(iii) (the replication phase), a template replicates. In step (iv) (the mixing phase), a strong external field $H^{'}$ is reoriented continuously to dissolve clusters. (e) A sample of the direction of the mixing field ${\hat{H}}^{'}$ indicated by the path it traces on the unit sphere.
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Figure 2
Dimer replication in the magnetic system. (a) Characteristic snapshots of the four steps of the replication cycle as they occur in simulations [18]. Black dots show the north pole of each dipole, and green dots show the binding sites. (b) The number of dimers $N_{d}$ over time for simulations with different values $τ$ . Dashed gray lines indicate best-fit models [Eq. (5)] of dimer counts for $τ = 25 τ_{0}$ and $τ = 100 τ_{0}$ . For small values of $τ$ the erroneous channel $R_{e}$ leads to entirely linear growth, while larger values begin to show nonlinearity. (c) The replication rate $R_{r}$ and the error rate $R_{e}$ in units of $τ_{0}^{- 1}$ for various values of $τ$ . Errors were calculated using the standard deviation of best fit values, which were found using a run-replacement bootstrap method (details in Appendix pp3).
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Figure 3
The two distributions governing replication efficiency for the magnetic system. (a) The number density of free monomers at time $t$ after the mixing cycle ends. The dotted line shows the free monomer density if the system begins in a truly random initial state, while the solid line shows the result if the system is randomized by the mixing field. Both results are averaged from 10 simulations of 500 monomers. (b) The binding probability for an isolated dimer that starts with two attracted monomers in the binding configuration at $t = 0$ , as in Fig. 1[d(ii)], determined by $10 000$ simulations of an isolated-dimer–two-monomer set. If the monomers remained in the binding configuration, then a simple exponential distribution would be observed (shown by the dotted line). In simulations (solid line), we observe a subexponential saturation as monomers transition from binding to trapping configurations.
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Figure 4
Replication with electrostatic repulsion. (a) An electrically charged dipole surrounded by a double-layer of counterions. We choose the layer thickness to be less than $σ$ but greater than $r_{b}$ . (b) The path traced on the unit sphere ${\hat{H}}^{'}$ in the low-error regime. The upper hemisphere shows the slow initial spiral to the $x - y$ plane, followed by motion similar to Fig. 1. (c) The correct binding configuration (right) next to the most commonly observed kinetic trap (left) for the case of a dimer with one (i) or two (ii) attracted monomers. (d) Growth of the number of dimers in the low-error regime with $τ = 300 τ_{0}$ and $q = 12$ for initial dimer counts of 0 and 10. Each solid line is the average of 20 simulations of 1000 total dipoles. With more than 60.6 dimers, replication is the dominant channel for dimer production. Growth falls below exponential as the system exhausts free monomers.
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Figure 5
A possible fabrication method, adapted from Ref. [27]. (a) Precursor colloids with a magnetite core coated by ${SiO}_{2}$ to a silicon surface. (b) The colloids are deposited on a silicon disk at low densities using a strong surfactant solution to prevent contact. Colloids will automatically assemble so the surface presented to the interface is orthogonal to the dipole direction. An external field will assist this process and help prevent multilayer aggregation. (c) The disk is spin coated with photoresist. (d) The photoresist is etched to slightly less than the colloid diameter. (e) Au is deposited on the exposed patches. (f) The photoresist is dissolved and colloids sonically removed and suspended in a solution with an appropriate surfactant. Colloids are finalized by self-complementary single-stranded DNA with an Au linker at one terminus, which adsorbs to the gold patch.
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Figure 6
Kinetic trapping. (a) The triangle trap family, shown with one and two particles. (b) The line trap family, shown with two or three particles. (c) Trapping energies normalized by the binding energies of the corresponding configuration. A value greater than 1 indicates that the trap is energetically favorable. In this plot, $H = 0.25 k$ .
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