Translational and rotational dynamics of colloidal particles interacting through reacting linkers

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Translational and rotational dynamics of colloidal particles interacting through reacting linkers

Pritam Kumar Jana and Bortolo Matteo Mognetti

Phys. Rev. E 100, 060601(R) – Published 2 December 2019

Abstract

Much work has studied effective interactions between micron-sized particles carrying linkers forming reversible, interparticle linkages. These studies allowed understanding the equilibrium properties of colloids interacting through ligand-receptor interactions. Nevertheless, understanding the kinetics of multivalent interactions remains an open problem. Here, we study how molecular details of the linkers, such as the reaction rates at which interparticle linkages form or break, affect the relative dynamics of pairs of cross-linked colloids. Using a simulation method tracking single binding and unbinding events between complementary linkers, we rationalize recent experiments and prove that particles' interfaces can move across each other while being cross-linked. We clarify how, starting from diffusing colloids, the dynamics become arrested when increasing the number of interparticle linkages or decreasing the reaction rates. Before getting arrested, particles diffuse through rolling motion. The ability to detect rolling motion will be useful to shed new light on host-pathogen interactions.

Received 17 May 2019

DOI:https://doi.org/10.1103/PhysRevE.100.060601

Physics Subject Headings (PhySH)

Biomimetic & bio-inspired materials Self-assembly

Grafted colloids

Coarse graining

Statistical Physics & ThermodynamicsPolymers & Soft MatterPhysics of Living Systems

Authors & Affiliations

Pritam Kumar Jana ^* and Bortolo Matteo Mognetti ^†

Center for Nonlinear Phenomena and Complex Systems, Code Postal 231, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium

^*Pritam.Kumar.Jana@ulb.ac.be
^†bmognett@ulb.ac.be

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Vol. 100, Iss. 6 — December 2019

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Images

Figure 1
The system and the modeling strategy. (a) We study micron-sized colloids decorated with $N_{b}$ linkers of length $L = 10 nm$ forming reversible, interparticle linkages. (b) We map linkers onto thin rods tipped by reacting sticky ends (circles). $k^{on}$ and $k^{off}$ are, respectively, the rates at which linkages form and break and are a function of the binding free energy of the sticky ends free in solution ( $Δ G_{0}$ ) and the configurational space available to bound and free linkages (respectively, $Ω_{k q}$ and $Ω_{m}, Ω_{p}$ ). $d_{q k}$ (similarly $d_{i j}$ ) is the distance between two bound tethering points ( $d_{q k} = | r_{k} - r_{q} |$ ) and $z$ the local distance between interfaces. (c) $e_{α}^{A}$ and $e_{α}^{B, A}$ ( $α = 1, 2, 3$ ) denote the internal reference frame, respectively, of particle $A$ and particle $B$ , the latter written using the internal reference of $A$ . $ϕ$ denotes the relative rotation between $B$ and $A$ (see the text and Ref. [39] for the definition) while $Δ r$ the lateral displacement of the center of mass of particle $B$ measured in $A$ reference frame. We also track the lateral displacement of the closest ( $c$ ) and farthest point ( $f$ ) of $B$ from $A$ as measured at the starting time.
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Figure 2
Translational and rotational diffusion dynamics. (a), (b) Lateral and rotational displacement (see Fig. 1) as a function of time for three different reaction rates $k_{0}$ [see Eq. (3)] and $〈 n_{b} 〉 = 126$ . The shaded regions represent the statistical errors calculated using 50 trajectories [55, 56]. (c), (d) Translational and rotational diffusion constants as a function of $k_{0}$ for $〈 n_{b} 〉 = 8, 25, 126$ corresponding to $β Δ G_{0} = - 6.4, - 4.4, - 3.3$ , respectively. In all these calculations $N_{b} = 5 \times 10^{4}$ . The vertical line corresponds to the typical binding rate of DNA sticky ends free in solution. The horizontal line in (c) represents the typical value of $D_{tran}$ at the melting transition measured in Ref. [29]. (e) At a given value of $Δ G_{0}$ , different values of $k_{0}$ do not alter the number of linkages.
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Figure 3
The rate at which linkages break, $k^{off}$ , controls the diffusivity of cross-linked interfaces. (a), (b) Systems with the same number of linkages, $〈 n_{b} 〉$ , and the same off rate, $k^{off}$ , feature the same translational and rotational displacement. We use $〈 n_{b} 〉 = 26 - 29$ and $k^{off} = 3.6 \times 10^{4} s^{- 1}$ corresponding to $k_{0} = 2.9 \times 10^{6} M^{- 1} s^{- 1}$ and $k_{0} = 55 \times 2.9 \times 10^{6} M^{- 1} s^{- 1}$ , respectively, for the system with $N_{p} = 10^{4}$ and $N_{p} = 5 \times 10^{4}$ linkers [see Eq. (3)]. The statistical analysis is based on 50 independent trajectories.
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Figure 4
The internal dynamics of cross-linked colloids. While changing the number of linkages, $〈 n_{b} 〉$ , and the kinetic rates, $k_{0}$ , colloids slide (a), roll (b), (c), or stop diffusing (d). Particles roll when the translational diffusion of a reference point in the contact region, $c$ , becomes much smaller than the diffusion of the center of mass, $C$ , or $f$ [see Fig. 1 for the definitions]. We calculate $D_{tran}$ by averaging over 50 trajectories. We also report single trajectories of $Δ r_{c}$ ( $c$ ), $Δ r$ ( $C$ ), and $Δ r_{f}$ (f) [see Fig. 1]. The employed values of $〈 n_{b} 〉$ and $k_{0}$ are (a) $k_{0} = 2.9 \times 10^{9} M^{- 1} s^{- 1}$ and $〈 n_{b} 〉 = 8$ , (b) $k_{0} = 2.9 \times 10^{9} M^{- 1} s^{- 1}$ and $〈 n_{b} 〉 = 126$ , (c) $k_{0} = 2.9 \times 10^{5} M^{- 1} s^{- 1}$ and $〈 n_{b} 〉 = 8$ , and (d) $k_{0} = 2.9 \times 10^{5} M^{- 1} s^{- 1}$ and $〈 n_{b} 〉 = 126$ .
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