Translational and Rotational Dynamics of Colloidal Particles Interacting through Reacting Linkers
Pritam Kumar Jana Bortolo Matteo Mognetti
cond-mat.soft cond-mat.stat-mech physics.bio-ph



Pritam.Kumar.Jana@ulb.ac.be 
Center for Nonlinear Phenomena and Complex Systems, Code Postal 231, Universite Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium

bmognett@ulb.ac.be 
Center for Nonlinear Phenomena and Complex Systems, Code Postal 231, Universite Libre de Bruxelles, Boulevard du Triomphe, 1050 Brussels, Belgium


Much work has studied effective interactions between micron–sized particles carrying linkers forming reversible, inter-particle 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 inter–particle linkages form/break, affect the relative dynamics of pairs of cross-linked colloids. Using a simulation method tracking single binding/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 inter-particle 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.



Translational and rotational dynamics of colloidal particles interacting through reacting linkers 
    Bortolo Matteo Mognetti
    Received: date / Accepted: date
==================================================================================================



Introduction. 
Understanding the dynamics of colloids interacting through ligand/receptor (or linker) molecules forming inter-particle linkages is a pivotal challenge in nanoscience. For instance, particles functionalized by reactive linkers are currently used to self-assemble crystals <cit.> or disordered structures <cit.> featuring sought properties and responsive behaviors. Secondly, ligand-receptor interactions are broadly employed by biology to regulate cell adhesion and inter–membrane trafficking <cit.>. The level of complexity and functionality achieved in these biological systems is out of the reach of current in vitro designs, partially because of a still incomplete understanding of multivalent interactions <cit.>. 


Many investigations have studied effective interactions between particles forming reversible linkages <cit.>. 
Effective interactions account for entropic terms (such as the avidity of the system) through ensemble averages over all possible linkages featured by the system at a given colloids' configuration and have been used to study systems with micron sized particles not directly accessible to Molecular Dynamics simulations <cit.>. However, effective interactions cannot quantify the kinetics of multivalent interactions which, often, is pivotal in controlling the outcome of an experiment. For instance, most of the experiments using DNA sticky–ends <cit.> tethered to micron–sized colloids tempt to self-assemble disordered aggregates <cit.> rather than crystalline structures. Arrested aggregates are the result of sticky interactions, as corroborated by dedicated experiments showing how the relative diffusion between DNA coated surfaces proceeds through hopping steps happening during intervals in which the surfaces are free from any inter–particle linkage <cit.>. Such finding implies that bound colloids diffuse only in a narrow range of temperatures close to the melting temperature. However, more recent experiments employing densely functionalized colloids have reported diffusive dynamics over broader temperature ranges with the possibility for the colloids to move against each other while being cross–linked <cit.>. 


In this work, we study the relative diffusion of two 1μm diameter colloids functionalized by up to 5· 10^4 ligands <cit.> using a numerical framework tracking the reaction dynamics of each linker. We find that particles' interfaces can move past each other while being bound when the number of inter-particle linkages is sufficiently low or the rates of forming/breaking linkages are sufficiently high. The latter finding rationalizes the results of Ref. <cit.> given that at higher coating densities the rates of breaking linkages at the melting transition are higher. The mobility of particles forming few linkages underlies a recent breakthrough reporting crystals of nanoparticles stabilized by smaller, conjugated nanoparticles roaming through the lattice <cit.>. 
We also show how particles feature different internal dynamics depending on the balance between the number of linkages and the value of the rates of reaction. In particular, while decreasing the latter, colloids first start rolling <cit.>, and then stop moving (arrested dynamics). Our findings will be useful to optimize the design of functionalized particles and will provide a valuable tool to study complex cell-pathogen interactions in biology <cit.>. While the present manuscript studies the dynamics of bound colloids, previous contributions have already clarified how, at low reaction rates, aggregation is limited by the diffusivity of the particles  <cit.>.


The model.
We consider two colloids with a diameter equal to 1μm <cit.> (Fig. <ref>a) each of which is decorated with N_p complementary linkers tipped by reactive sites (full and open circles in Fig. <ref>b).  We map each linker (labeled with i, i=1,⋯ ,N_p) into a thin, rigid rod of length L=10nm free to pivot around its tethering point r_i (Fig. <ref>b). This representation provides analytic expressions of the force and torque acting on the colloids. Existing literature (e.g. <cit.>) has extensively used thin rods to model linkers made of double-stranded DNA terminated by a short single-stranded DNA oligomer (the reactive site, see Fig. <ref>b). The tethering points are uniformly distributed over the spheres' surfaces and do not move. We average our results over more than 50 different coating realizations 
sampled from independent, uniform distributions. Notice that for the largest employed values of N_p (N_p=5· 10^4),  the average distance between two neighbouring tethering points (which is equal to √(4 π R^2/N_p) =7.9nm) is large enough to neglect non-specific (e.g. electrostatic or steric) interactions between linkers.  


The timescales regulating colloidal dynamics at the molecular scale are the rates at which complementary linkers bind and unbind (k^on_ij and k^off in Fig. <ref>b). Coherently with what done in recent studies modeling DNA-mediated interactions <cit.>, we assume that the rates of breaking a linkage is not a function of the position of the tethering points and is equal to the denaturation rate of paired sticky ends when free in solution. The fact that at the thermal scale the forces exerted by the tethering points onto paired sticky ends are not sufficiently strong to deform the structure of the double helix <cit.> justifies the choice of using configuration independent off rates. This result does not apply to other types of ligand-receptor complexes <cit.>. Given k^off, we calculate k^on_ij from the hybridization free energy, Δ G^hyb_ij, using k^on_ij = k^offexp[-βΔ G^hyb_ij], where β=1/(k_BT), and T and k_B are the temperature and the Boltzmann constant, respectively. Δ G^hyb_ij comprises the hybridization free energy of free sticky ends in solution, Δ G_0 <cit.>, and a configurational part Δ G^conf=k_B T log ( Ω_ij/ ρ_0 Ω_i ·Ω_j ), where Ω_ij and Ω_i ·Ω_j are, respectively,  the configurational space available to linker i and j when bound, and when free. 
If d_ij and z_i are, respectively,  the distance between the tethering points of two bound linkers and between a tethering point of a free linker and the facing interface (see Fig. <ref>b), we have <cit.>

    Ω_ij = 2 π L^2  d_ijχ           Ω_i = 2 π L ·min[z_i,L].

Ω_ij is proportional to the length of the spherical arc traced by the reacted tips of the linkers times a Jacobian term while χ is the fraction of this spherical arc not excluded by the walls of the colloids. Ref. <cit.> reports the explicit expression for χ. To obtain portable expressions, we employ a mean field simplification in which we calculate χ by averaging over the possible lateral distances between tethering points. χ then reads as follows 

    χ (z) = {[               1 if   L< z < 2 L;      z  2 L - z      if   z < L ].

where z=(z_i+z_j)/2. Notice that after averaging over the lateral positions of the tethering points, the reacted tips are uniformly distributed along the direction orthogonal to the interfaces. χ is then the fraction of tips-to-plane distances not excluded by the colloids.
The final expression for Ω_ij follows from Eq. <ref> and Eq. <ref>.
The  on and off rates read as follows <cit.>

    k^off = exp[βΔ G_0] ρ_0 k^_0
                
    k^on_ij = Ω_ij/Ω_iΩ_j k^_0

where k_0 is the binding rate of free sticky ends in solution and we set ρ_0 to the standard concentration, ρ_0=0.62·nm^-3.
In this work, we use k_0 as a parameter to change the speed of the reactions without affecting equilibrium quantities (e.g., the  number of linkages). Our results are then not specific to DNA linkers but apply to other complexes featuring different k_0 <cit.>. The methodology could also be adapted to include possible cooperativities between binding events as characterized in Molecular Dynamics simulations <cit.>. 


We define by f^(A)_⟨ ij ⟩ the effective force exerted on colloid A by the linkage formed by linker i and j, labeled with ⟨ ij ⟩ (f^(B)_⟨ ij ⟩=- f^(A)_⟨ ij⟩). f^(A)_⟨ ij⟩ accounts for variations of the single linkage partition function, Ω_ij, and reads as f^(A)_⟨ ij⟩=k_B T ∇_ r_AlogΩ_ij, where r_A is the center of mass of colloid A. Using Eqs. <ref>, <ref> we find

    β f^(A)_⟨ ij ⟩   =   {[                           -û_ d_ij  d_ij                          if   L< z < 2 L; -û_ d_ij  d_ij +  û_z ( 1  z
    + 1  2L -z)                               if   z < L ].

where û_ d_ij and û_z are the unit vectors, respectively, pointing in the direction of d_ij and orthogonal to the surface of the colloids. For large particles (R/L≫ 1), û_z is approximated by the center-to-center unit vector û_z = ( r_A -  r_B)/| r_A -  r_B|. Similarly, each unbound linker i (either tethered to A or B colloid), labeled with ⟨ i ⟩, exerts a force on colloid A equal to f^(A)_⟨ i ⟩=k_B T ∇_ r_AlogΩ_i which reads as follows

    β f^(A)_⟨ i ⟩   =   {[               0 if   L< z < 2 L;        û_z  z_i      if   z < L ].

(a similar expression follows for β f^(B)_⟨ i ⟩).
Notice how f^(A)_⟨ ij ⟩ comprises an attractive contribution along û_ d_ij and a repulsive term along the centers of mass direction û_z due to entropic compression. f^(A)_⟨ i ⟩ only includes a repulsive term along û_z. 
The hard-core interaction between colloids is modeled using the repulsive interaction engendered by a uniform distribution of inert linkers of length 0.75· L and reads as follows 

    β f^(A)_rep   =    2πRσlog(0.75L/D) û_z

where σ and D are, respectively, the density of the inert linkers (σ=0.003nm^-2) and the face-to-face distance (D=| r_A- r_B|-2R). A similar expression follows for f^(B)_rep. Notice that Eq. <ref> follows from a Derjaguin approximation and Eq. <ref>. Inert linkers are often used in experiments to stabilize colloidal suspensions. 


The total force, f^(A), and torque, τ^(A), acting on particle A are then equal to 𝐟^(A)=∑_<ij>𝐟_<ij>^(A)+∑_<i>𝐟_<i>^(A) and τ^(A)=∑_<ij> (𝐫_i-𝐫_A) ×𝐟_<ij>^(A), where the sums are taken over the list of linkages and free linkers. 
Notice that Δ G^hyb_ij and f^(A) are not a function of the reference density ρ_0.
We simulate changes in the position, Δ r_A, and orientation, Δφ_A, of the colloids using a Brownian Dynamics scheme <cit.> (similar equations hold for Δ r_B and Δφ_B):

    Δ r_A=𝐟^(A) D^T Δ t k_BT + δ𝐫 ;
             Δφ_A=τ^(A) D^RΔ t k_BT+δφ

We sample δ𝐫 and δφ from a Gaussian distribution with zero mean and variance equal to, respectively, 2D^TΔ t and 2D^RΔ t, where Δ t is the integration time step (Δ t=2ns). D^T and D^R are the translational and rotational diffusion coefficients given by the Stokes–Einstein–Debye relations: 

    D^T=k_BT/6πηR, 
         
    D^R=k_BT/8πηR^3,

where k_B is the Boltzmann constant, T the temperature (T=298 K), and η the dynamic viscosity (η=0.91 mPa· s). These figures lead to D^T=4.7·10^5nm^2s^-1 and D^R=1.4  rad^2s^-1. 


At each step of the simulation dynamics, we first update the list of linkages ⟨ ij ⟩ using the Gillespie algorithm <cit.> as done in Refs. <cit.>. 
At a given colloid configuration, { r_A,   e^A_α,  r_B,   e^B_α}, we start calculating all the on/off rates  of making/breaking linkages between all possible linkers, k^on_ij and k^off_ij,  as derived in Eq. <ref>. We then calculate the affinity of a reaction to happens as

    a_on^ij= δ_⟨ i⟩δ_⟨ j⟩ k_on^ij, 
       
     a_off^ij=δ_⟨ ij⟩ k_off^ij,

where δ_⟨ i⟩=1 (δ_⟨ j⟩=1) if linker i (j) is free (δ_⟨ i⟩=δ_⟨ j⟩=0 otherwise), and δ_⟨ ij⟩=1 if a linkage between i and j is present (δ_⟨ ij⟩=0 otherwise). Notice that we treat each linker as an independent chemical species.  

We then fire one within all possible reactions with probability 

    p_on^ij= a_on^ij a_tot, 
       
     p_off^ij= a_off^ij a_tot,

where a_tot is the sum of all affinities. Along with choosing one possible reaction, we sample the time for it to happen (τ), distributed as P(τ) =a_totexp[-a_totτ], and increment a reaction clock τ_reac by τ. If τ_reac< Δ t (where Δ t is the simulation step), we update δ_⟨ i⟩ and δ_⟨ ij⟩, recalculate the affinities (Eq. <ref>), and fire a new reaction until reaching Δ t. At that point we calculate forces using Eqs. <ref>, <ref>, <ref> and update { r_A,   e^A_α,  r_B,   e^B_α} using  Eq. <ref>. The choice of firing more than a single reaction between consecutive configurational updates was motivated by the need of having an affordable algorithm at high reaction rates. Understanding potential bias arising from this limit deserves future dedicated investigations. As compared to existing methods <cit.>, our algorithm allows simulating configurations in which linkers can potentially bind multiple partners. We neglect hydrodynamic interactions consistently with the fact that the latter are not considered when modeling the linker dynamics. 



Analysis. We define by e^A_α(t) and e^B_α(t) (α=1, 2, 3) the internal reference frame, respectively, of particle A and B. To study the relative motion between colloids, we consider the dynamics of the internal reference frame of particle B in the reference frame of particle A, e^B,A_α(t), e^B,A_α(t)= O e^B_α(t) (see Fig. <ref>c). O is a rotation matrix with line i given by the transpose of e_i^A(t). We then calculate the translational diffusion constant, D_tran, by studying the lateral displacement of the center of mass of B (Δ r in Fig. <ref>c). Instead, we calculate the rotational diffusion constant D_rot from ϕ(t) defined as ϕ(t)=∑_t<t w (t) with w (t)=  e^B,A_α(t) × e^B,A_α(t+Δ t) at a given α <cit.>. To quantify sliding versus rolling motion, we also study the lateral displacement of the closest, c, and furthermost, f, points of B from A (Δ r_c and Δ r_f in Fig. <ref>c). We define the translational and rotational diffusion coefficients as ⟨Δ r^2⟩=4D_trant and ⟨ϕ^2⟩=4D_rott, respectively.


Results and Discussion. 
In Fig. <ref>a-d, we study D_tran and D_rot as a function of the reaction rate, k_0, and the average number of linkages, ⟨ n_b ⟩. As predicted by Eq. <ref>, different values of k_0 do not alter the binding free energy, Δ G^hyb_ij=-k_B T log(k^on_ij/k^off), and therefore the  average number of linkages ⟨ n_b ⟩ (Fig. <ref>e). In Fig. <ref>a and b we focus, respectively, on ⟨Δ r^2 ⟩ and ⟨ϕ^2 ⟩ at three different values of k_0 for systems with ⟨ n_b ⟩=126. We find a diffusive trend for the two highest values of k_0 while for the smallest one the dynamics are arrested. Fig. <ref>c and <ref>d reports, respectively, D_tran and D_rot for three different values of ⟨ n_b ⟩ (corresponding to three different values of Δ G_0) and k_0 spanning 10 orders of magnitudes. 

Error bars a calculated as in Refs. <cit.>. 
Overall the diffusion coefficients increase with k_0 and decrease with ⟨ n_b ⟩. We observe arrested dynamics only when ⟨ n_b ⟩=126 and k_0<10^8M^-1s^-1. Our results show how stickiness between particles arises from having many persistent linkages. Instead, when linkages are often reconfigured (at high k_0), the interfaces can move past each other even when ⟨ n_b ⟩ > 0. Below, we clarify how the linkages couple the rotational and translational motion of the particles resulting in rolling dynamics.  

The phenomenology of Fig. <ref> is consistent with the experimental results of Ref. <cit.> studying the relative motion of two bound colloids (one of which is kept fixed) at the melting transition. Well below the melting temperature (corresponding to high values of ⟨ n_b ⟩) the motion is subdiffusive. Instead, close to the melting temperature (small values of ⟨ n_b ⟩) Ref. <cit.> reported diffusive motion with D_tran comparable to the horizontal magenta line in Fig. <ref>a (corresponding to D_tran=0.2 μm^2s^-1). A sound estimation of k_0 for short DNA oligomers is k_0=10^6M^-1s^-1 <cit.> (see the vertical line in Fig. <ref>c and <ref>d). The experimental value of D_tran corresponds to systems with around 10^1 linkages as usually found at the melting transition <cit.>. The successful comparison between simulations and experiments corroborates our model that will be useful to study the rotational motion of the particles which cannot be studied in experiments (see Fig. <ref>d and <ref>).



The results of Fig. <ref> prove the pivotal role played by k_0, along with the number of bridges ⟨ n_b ⟩, in controlling the dynamics of the system. We now clarify how the single bridge lifetime, (k^off)^-1, is the only timescale of the model that affects colloids’ diffusion (at a given ⟨ n_b ⟩). To do so, in Fig. <ref> we compare two systems with a total number of linkers per particle equal to N_p= 10^4 and N_p=5· 10^4. 

These two systems feature coating densities comparable with the ones found in two classes of experiments employing different tethering molecules, Ref. <cit.> and Ref. <cit.>. We fine-tune the values of Δ G_0 in such a way that the numbers of bridges displayed by the two pairs of particles remain comparable, ⟨ n_b ⟩≈ 26-29. In particular, for the N_p= 10^4 and N_p=5· 10^4 case, we use, respectively,  βΔ G_0=-8.4 and βΔ G_0=-4.4. As expected, bridges are stronger (Δ G_0 is lower) for N_p=10^4 than for N_p=5· 10^4 due to combinatorial entropy terms favoring bridge formation in the latter case. Given the two values of Δ G_0, we calculate the corresponding values of k_0 (see caption of Fig. <ref>) leading to the same k^off as predicted by Eq. <ref>. We then simulate the diffusion dynamics of the two systems (Fig. <ref>). Remarkably, Fig. <ref> shows how the two systems feature identical rotational and translational dynamics. 





We now study how the linkages constrain the relative dynamics of the two particles resulting in a coupling between the rotational and the translational motion of the colloids. In turn, this coupling enhances rolling dynamics over sliding <cit.>. A colloid slides if its internal reference frame remains parallel to the tangential plane of the opposing surface, while rolls if the closest point to the facing surface, c in Fig. <ref>c, remains at rest during an update. Therefore in Fig. <ref> we compare the lateral displacement of point c with the one of point f and the center of mass C (see Fig. <ref>c). Because the position of the contact point does not change significantly during our simulations, we fix c (and similarly f) to the contact point at time t=0 (see Fig. <ref>c). 
The diffusion of the contact point c is maximized at low values of ⟨ n_b ⟩ and high reaction rates, k_0, (see Fig. <ref>a). In such conditions, colloids slide. When increasing the number of bridges (Fig. <ref>b) or decreasing the reaction rate (Fig. <ref>c) the diffusion of the point c drastically slow down, and colloids start rolling. Intriguingly, the diffusion constants of Figs. <ref>b and <ref>c are comparable. Finally, at low k_0 and high ⟨ n_b ⟩, the dynamics are arrested (Fig. <ref>d). A recent theoretical contribution also predicted rolling motion <cit.>. Unfortunately, Ref. <cit.> used a simplified model in which linkers bind any point of the facing surface (instead of complementary partners) hampering a direct comparison with our results.  



Conclusions. In this work we have studied the internal dynamics of colloids decorated by linkers with fixed tethering points forming reversible inter-particle linkages. We have clarified how the average number of linkages, ⟨ n_b ⟩, and the rates at which linkages form and break, k_0 in Eq. <ref>, are the key parameters controlling the relative diffusion of cross-linked particles. We confirm the results of recent experiments employing DNA linkers showing how cross-linked colloids can diffuse while being bound <cit.>. We explain how this result is due to the higher coating density employed by Ref. <cit.> than what used in previous studies resulting in faster reaction kinetics at the melting transition. We also clarify how the balance between the number of linkages ⟨ n_b ⟩ and reaction rates k_0 leads to rolling motions.  Our results and quantitative methodology will be useful to study biological systems, like virus infection, where the number of linkages between host and pathogen is tightly regulated (e.g. <cit.>).


Acknowledgement. 
This work was supported by the Fonds de la Recherche
Scientifique de Belgique (F.R.S.)-FNRS under grant n^∘ MIS
F.4534.17. Computational resources have been provided
by the Consortium des Équipements de Calcul Intensif 
(CECI), funded by the F.R.S.-FNRS under grant n^∘ 2.5020.11. 


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