Bond Flexibility and Low Valence Promote Finite Clusters of Self-Aggregating Particles

Olga Markova, Jonathan Alberts, Edwin Munro, and Pierre-François Lenne

Phys. Rev. Lett. 109, 078101 – Published 13 August 2012

Abstract

Systems of complex particles such as proteins or colloidal particles have a widely observed tendency to form nonconnected nanometer-size clusters at steady state, but the underlying mechanisms remain poorly understood. We report here a numerical study on the self-aggregation of low-valence particles with flexible bonds (i.e., free bond orientations) in two dimensions and predict the formation of a stable cluster phase for average valences ranging from 2 to 3.6. For the intermediate case of trivalent particles, we show that a cluster phase is present over a wide range of concentrations and interaction energies. The clusters are polydisperse in size, have a fractal dimension of 1.5, and tend to fully saturate their bonds at high interaction energies. The number of unformed bonds scales linearly with the number of particles in a cluster, which implies the absence of phase transition in the explored region of interaction energies and concentrations. We discuss possible implications of our model for membrane protein clustering.

Received 11 November 2011

DOI:https://doi.org/10.1103/PhysRevLett.109.078101

Authors & Affiliations

Olga Markova¹, Jonathan Alberts², Edwin Munro^3,*, and Pierre-François Lenne^1,†

¹Institut de Biologie du Développement de Marseille-Luminy, UMR7288 CNRS/Aix-Marseille Université, Campus de Luminy, 13288 Marseille Cedex 9, France
²University of Washington, 620 University Road, Friday Harbor, Washington 98250, USA
³Department of Molecular Genetics and Cell Biology, University of Chicago, 5801 South Ellis Avenue, Chicago, Illinois 60637, USA

^*emunro@uchicago.edu
^†pierre-francois.lenne@univ-amu.fr

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Issue

Vol. 109, Iss. 7 — 17 August 2012

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Images

Figure 1
Snapshot of a simulation of 5000 particles with $valence = 3$ at steady state. Particles that were initially randomly distributed self-organize into heterogeneous-in-size dynamical clusters at steady state. Surface concentration is 0.06; $ε = 6.8 kT$ . Inset: zoom view of one cluster. Color of particles indicate the number of its current partners: pink (or dark gray in the gray-scale figure), 1; black, 2; cyan (or light gray in the gray-scale figure), 3. The painted radius of particles is 80% of the actual radius to facilitate visualization of bond orientations. $d_{f} = 1.5$ denotes the fractal dimension of clusters.Reuse & Permissions
Figure 2
Analysis of steady-state cluster size distributions. Clusters were defined based on particle connectivity; thus, particles belong to the same cluster if they are connected by bonds directly or through intermediate particles. (a) Cluster size distribution for different bond energies $ε$ indicated on the graph. (b) Dependence of mean cluster size on bond energy. (c) Cluster size distribution at several concentrations indicated on the graph. (d) Dependence of mean cluster size on the concentration.Reuse & Permissions
Figure 3
Dependence of the number of unformed bonds per cluster on cluster size for valences 2–6. The valence is marked near each curve. In addition, the case of three regularly fixed bonds is presented (marked 3p) for direct comparison to the case of freely oriented bonds. Interaction energies for a trivalent system (both free and fixed bonds) $ε = 7 kT$ . For each curve, $0.5 - 1 \times 10^{6}$ clusters were analyzed. Inset: slope of dependence of the number of unformed bonds on cluster size in log-log scale. For valence 3, the red symbol represents the fixed bonds model (slope $= 0.5$ ), and the violet symbol represents the free bond model (slope $= 1$ ). The slope shows the square root dependence for the number of unformed bonds on cluster size for valence 4–6 and fixed bonds model and linear dependence for free bond orientation trivalent particles.Reuse & Permissions
Figure 4
Dependence of bond saturation on interaction energy for trivalent clusters with flexible bonds. $V = 3$ . (a) Dependence of the number of unformed bonds per cluster on cluster size at different interaction energies marked on the graph; $10^{5}$ clusters were analyzed for each interaction energy. (b),(c) dependence of fitting coefficients $a$ and $b$ on the interaction energy $ε$ : $a = 0.61 e^{- ε / 2.5}$ ; $b = 1 - 0.06 ε$ for the range of $ε \in [2 ∶ 22] kT$ .Reuse & Permissions
Figure 5
Cluster size distribution for mixtures with average valences ranging from 2 to 4 at steady state. The binary mixtures of 2–3 and 3–4 valence particles set the effective valence. Interaction energy is 3.8 kT for valency 3–4 and 8 kT for valences 2 and 2.4. For better visualization, the curves have been scaled by a constant.Reuse & Permissions

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