Random walk of passive tracers among randomly moving obstacles
 Matteo Gori^{1, 2},
 Irene Donato^{1, 2},
 Elena Floriani^{1, 2},
 Ilaria Nardecchia^{2, 3} and
 Marco Pettini^{1, 2}Email author
https://doi.org/10.1186/s1297601600381
© Gori et al. 2016
Received: 10 December 2015
Accepted: 19 March 2016
Published: 14 April 2016
Abstract
Background
This study is mainly motivated by the need of understanding how the diffusion behavior of a biomolecule (or even of a larger object) is affected by other moving macromolecules, organelles, and so on, inside a living cell, whence the possibility of understanding whether or not a randomly walking biomolecule is also subject to a longrange force field driving it to its target.
Method
By means of the Continuous Time Random Walk (CTRW) technique the topic of random walk in random environment is here considered in the case of a passively diffusing particle among randomly moving and interacting obstacles.
Results
The relevant physical quantity which is worked out is the diffusion coefficient of the passive tracer which is computed as a function of the average interobstacles distance.
Conclusions
The results reported here suggest that if a biomolecule, let us call it a test molecule, moves towards its target in the presence of other independently interacting molecules, its motion can be considerably slowed down.
Keywords
Background
The topic of random walk in random environment (RWRE) has been the object of extensive studies during the last four decades and is of great interest to mathematics, physics and several applications. There is a huge literature on numerical, theoretical, and rigorous analytical results. The subject has been pioneered both through applications, as is the case of the models introduced to describe DNA replication [1], or through more abstract models in the field of probability theory [2]. One can find in Ref. [3] the definition of the mathematical framework of RWRE and since then a vast body of results has been built for both static and dynamic random environments, to mention just a few of them see [4–7] and the references therein quoted.
An example of biophysical application of RWRE is related with singleparticle tracking experiments allowing to measure the diffusion coefficient of an individual particle (protein or lipid) on the cell surface; the knowledge of singletrajectory diffusion coefficient is useful as a measure of the heterogeneity of the cell membrane and requires to model hindered diffusion conditions [8].
To give another example among a huge number of processes in living matter, during B lymphocyte development, immunoglobulin heavychain variable, diversity, and joining segments assemble to generate a diverse antigen receptor repertoire. Spatial confinement related with diffusion hindrance from the surrounding network of proteins and chromatin fibres is the dominant parameter that determines the frequency of encounters of the above mentioned segments. When these particles encounter obstacles present at high concentration, the particles motions become subdiffusive [9] as described by the continuous time random walk (CTRW) model [8, 10].
In a biophysical context this kind of problems is referred to as “macromolecular crowding” which, among other issues, encompasses the effects of excluded volume on molecular diffusion and biochemical reaction rates within living cells. This problem has been largely studied both experimentally and numerically over the years (see respectively [11, 12] and references therein).
In this paper, we consider a very simplified model in order to obtain analytical results on the diffusion coefficient of passive tracers evolving among interacting and randomly moving particles. The prospective reason for studying this problem stems from the need of estimating how the encounter time of a given macromolecule (passive tracer) with its cognate partner, say a transcription factor diffusing towards is target on the DNA, is affected by the surrounding particles intervening in other biochemical reactions.
The complexity of real crowded systems appears at the moment very difficult to be managed by analytical calculations, for these reasons we have made important simplifications with respect to the realistic case. In particular, we have limited our analysis to a low concentration limit for the obstacles, assuming that the average distances among the particles (both tracers and obstacles) is much larger than their characteristic dimensions. Although this assumption is unrealistic in vivo, the present work can be considered as a first step in a feasibility study for an experiment oriented to infer whether intermolecular electrodynamic long range forces are at work in living matter using dilute solutions of biomolecules in vitro. This is in the same line as some recent works ([13–16]).
Methods: continuous time random walk formalism
One of the many ways of modelling diffusive behavior is by Continuous Time Random Walk (CTRW) [17, 18]. This framework is mainly used to extend the description of Brownian motion to anomalous transport, in order to deal with subdiffusive or superdiffusive behavior in connection with Lévy processes, but it can of course be used to describe the simpler and more frequent case of normal diffusion. In this paper, we focus on cases where diffusion of tracers and interacting molecules is indeed Gaussian, so that a diffusion coefficient can be defined.

The Velocity Model, in which each particle A moves with constant velocity v _{0} between two turning points; at a turning point, a new direction and a new length of flight are taken according to the probability density Λ(r).

The Jump Model, in which each particle waits at a particular location before instantaneously moving to the next one, the displacement being chosen according to the probability density Λ(r), the waiting time for a jump to take place being r/v _{0}.
The expression of P(r,t) is formally different for these two versions of the CTRW, but from their definition it appears that the two models are equivalent in the long time limit.
As a general remark on other possible applications of our work, this CTRW description where space and time are coupled (see Eq. (3)) allows us to model situations not only of Gaussian diffusion but also of enhanced diffusion (where 〈r ^{2}(t)〉≃t ^{ α } with α>1) [18], because it can describe cases where the particles keep the same velocity for very long times (if the freeflight distribution ϕ(t) decays slowly, typically as an inverse power law).
Let us notice that the same CTRW formalism can also describe subdiffusion (where 〈r ^{2}(t)〉≃t ^{ α } with α<1) [18]. This can be obtained by considering a version of the Jump Model where space and time are decoupled, as in Eq. (2): particles remain at a particular location for times distributed according to ψ(t) and make instantaneous jumps on distances distributed according to Λ(r). Subdiffusion is obtained as soon as Λ(r) has finite second moment while the first moment of the waiting time distribution ψ(t) diverges.
Results and discussion
Diffusion of independent tracers in the presence of interacting obstacles
If we adopt the CTRW description of diffusion presented in the preceding section, then the main quantity to consider is ϕ _{ A }(t), the probability density function that a random walker A keeps the same direction of velocity during a time t.
where R _{ A } is the hydrodynamic radius of the diffusing particles and η is the viscosity of the medium where the particles diffuse.
where m _{ A } is the mass of a particle A.
As stated in the introduction, the physical situation we are interested in is the one where another population of particles, say Bparticles, is also present in the solution. Particles B are supposed to diffuse and mutually interact, but there is no interaction at a distance between them and the particles A. It is reasonable to suppose that the diffusive and dynamic properties of these moving obstacles B induce changes in the diffusive properties of the Aparticles which can be thus seen as passive tracers.
We want to model how the Bparticles affect the diffusion properties of the Aparticles by resorting to a suitable modification of the CTRW probability distribution ϕ _{ A }(t). The amount of the modification will of course depend on the concentration C _{ B } (or equivalently on the average distance \(d=C_{B}^{1/3}\)) of obstacles. Our goal is to estimate with simple arguments the dependence on the average distance d between any pair of obstacles of the ratio \(D_{A}/D_{0_{A}}\phantom {\dot {i}\!}\) between perturbed and unperturbed diffusion coefficients.
so that the Aparticles can be regarded as tracers: any Aparticle does not influence the dynamics of the obstacles and of the other tracers.
Modification of the microscopic freeflight time distribution
we can consider that the diffusion of Aparticles is not perturbed by the presence of the obstacles B; thus for the waiting time distribution we will have \(\phantom {\dot {i}\!}\phi _{A}(t)\simeq \phi _{0_{A}}(t)\), and, consequently, \(\phantom {\dot {i}\!}D_{A}\simeq D_{0_{A}}\).
As the concentration of Bparticles grows, the diffusion of Aparticles is affected accordingly, and this is described by a modification of ϕ _{ A }(t). It is reasonable to suppose that ϕ _{ A }(t) will be close to \(\phi _{0_{A}}(t)\) at sufficiently short times, i.e., for displacements small enough that a tracer A does not “see” any obstacle B, and that ϕ _{ A }(t) will be reduced with respect to the unperturbed \(\phi _{0_{A}}(t)\) at long times, because long free displacements are likely to be interrupted by the presence of obstacles.
Following this idea, we model the waiting time distribution as follows: we call T _{ d } the characteristic time of flight at which a tracer A begins to “see” the obstacles B, where T _{ d } depends of course on the typical distance \(d\simeq C_{B}^{1/3}\) between the Bparticles. We then make the simplest assumption that ϕ _{ A }(t) coincides (except for a normalisation factor) with \(\phi _{0_{A}}(t)\) for times smaller than T _{ d } and is zero for times larger than T _{ d }.
which is a function of the ratio between the transition time T _{ d } and the characteristic timescale τ _{ A } of the non perturbed waiting time distribution. The issue is now to establish the dependence of the transition time T _{ d } (and consequently, of the parameter x) on the average distance d between obstacles.
The fact that the obstacles move under the influence of deterministic nonlinear interparticle potentials implies a chaotic dynamics which apriori could be very different from a stochastic dynamics, this notwithstanding such a chaotic dynamics entails a Brownianlike diffusion as was found by numerical simulations in Ref. [14]. Hence we assume that the Bmolecules (obstacles) diffuse with Brownian motion: we can apply to them the CTRW description with velocity \(v_{0_{B}}\) and waiting time distribution ϕ _{ B }(t), corresponding to a situation where they do not interact. We can then approximately take into account their mutual interaction by giving them a systematic drift velocity that is due to deterministic forces acting between them. This drift velocity depends on their mutual distance d, and we will call it V _{ d }. If we suppose that the dynamics of the Bmolecules is overdamped, a crude estimation of V _{ d } is given by V _{ d }≃F(d)/γ _{ B }, where γ _{ B }=6π R _{ B } η is the friction coefficient of the Bmolecules and F(d) the norm of the deterministic force between two molecules of type B at a distance \(d=C_{B}^{1/3}\).
where we have used Eqs. (16), (17) and (20).
Equation (24) is a rough estimate of this characteristic time because it excludes, for instance, effects due to the dimensionality of physical space where diffusion takes place (1D, 2D, etc.), the sign of interaction energy among obstacles, spatial correlation among obstacles and the possibility of multiple collisions among the molecules. The last point entails the exclusion  from the range of validity of our model  of all the cases where d≲ min{R _{ A },R _{ B }} (as in the case of densely crowded systems). For this reason we do not take into account the sizes of both tracers and obstacles at a distance d from the colliding particle.
where q is the elementary charge expressed in Gaussian units. Using (24), the transition time is T _{ d }≃3·10^{−4} μs, whence we get x(d)≃6·10^{3}.
Modification of the rescaled freeflight time distribution
In order to describe physical systems for which T _{ d }≫τ for all the accessible values of the intermolecular distance d, as the one described by the preceding example, we have to modify the CTRW model.
where, analogously to the previous case, \(\tilde {v}_{0_{B}}=\alpha _{B} v_{0_{B}}\) and \(\tilde {\tau }_{B}=\beta _{B}\tau _{B}\).
Of course this does not model the microscopic level, in the sense that the single motional events  whose probability is specified by \(\tilde {\psi }_{A} (\textbf {r},t)\)  are no longer the microscopic displacements between successive Brownian collisions. Rather, we focus on the motion on longer timescales \(\tilde {\tau }_{A}\) (β _{ A }>1) and model the diffusion of tracers as a sequence of displacements on typical distances \(\tilde {v}_{A_{0}}\tilde {\tau }_{A}\).

the typical motional event for tracers (Aparticles) takes place between two consecutive encounters with an obstacle (Bparticles); this means that the spatial scale of a typical motional event for tracers described by \(\tilde {\psi }_{0_{A}}(\textbf {r},t)\) is d, the average distance between any two obstacles. This condition guarantees that τ _{ A }, and consequently \(\tilde {\psi }_{0_{A}}(\textbf {r},t)\), is modified in the presence of obstacles:$$ \left(\tilde{v}_{0_{A}}+\tilde{v}_{0_{B}}\right)\tilde{\tau}_{A} = \left(\alpha_{A} v_{0_{A}}+\alpha_{B} v_{0_{B}}\right)\beta_{A} \tau_{A}= d $$(30)

for Bparticles we can also write a condition analogous to Eq. (30) under the assumption that the motional events for obstacles are determined by encounters among them in absence of mutual interactions. This is justified by the assumption that the concentration of tracers is negligible compared with the concentration of obstacles. In this framework it is reasonable to assume:$$ 2\tilde{v}_{0_{B}}\tilde{\tau}_{B}=2\alpha_{B} \beta_{B} \left(v_{0_{B}}\tau_{B}\right)=d $$(31)

the dynamics of tracers is now dominated by the encounters with obstacles, that means$$ \frac{\tilde{v}_{0_{A}}^{2}\tilde{\tau_{A}}}{3}=D_{exVol_{A}}(d) $$(32)where \(D_{exVol_{A}}(d)\) is the diffusion coefficient of tracers taking into account the excluded volume effects due to the presence of the obstacles. As we are investigating the case d≫R _{ A }+R _{ B }, we can neglect the excluded volume effects and substitute \(D_{0_{A}}=D_{exVol_{A}}(\infty)\), yielding:$$ \frac{\tilde{v}_{0_{A}}^{2}\tilde{\tau_{A}}}{3} = \frac{{\alpha_{A}^{2}} \beta_{A} \left(v_{0_{A}}^{2}\tau_{A}\right)}{3} = \frac{v_{0_{A}}^{2}\tau_{A}}{3}=D_{0_{A}} \;\;\;\; \Rightarrow \;\;\;\; {\alpha_{A}^{2}}\beta_{A} =1 \ $$(33)

the considerations in the previous item can be extended to obstacles (Bparticles) if no interactions act among them, so that:$$ \frac{\tilde{v}_{0_{B}}^{2}\tilde{\tau_{B}}}{3} = \frac{{\alpha^{2}_{B}}\beta_{B}\left(v_{0_{B}}^{2}\tau_{B}\right)}{3} = \frac{v_{0_{B}}^{2}\tau_{B}}{3}=D_{0_{B}} \;\;\;\; \Rightarrow \;\;\;\; {\alpha_{B}^{2}}\beta_{B} =1 $$(34)
Notice that the rescaled velocity and time now implicitly depend on the parameter d.
where, as α _{ A }>0, the physical solution we choose is the one with the “ + ” sign.
Here V _{ d } is the drift velocity of the obstacles, that we can estimate in the same way as in Section “Modification of the microscopic freeflight time distribution”, that is, V _{ d }≃F(d)/γ _{ B }. For both conditions, it is evident that \(\tilde {T}_{d}\le \tilde {\tau }_{A}\), where the equality holds when V _{ d }=0, that is, the Bparticles do not interact.
where we have used Eq. (38) for β _{ A }.
Slowing down of Brownian diffusion: the patterns of D/D _{0}
In this section we report the patterns of the ratio \(D_{A}/D_{0_{A}}\phantom {\dot {i}\!}\) obtained by means of the theoretical expressions (23), (25) and (41), (42). We denote by D and D _{0} the diffusion coefficients of the tracers (Aparticles) in the presence and in the absence of obstacles (Bparticles), respectively. We plot this ratio as a function of the average distance d between any two obstacles obtained for different kinds of interaction potentials between the Bparticles: screened electrostatic potential, Coulombic potential, dipolar potential. These potentials have been chosen as they are representative of some relevant interaction in biology [19]. The choice of Coulombic and dipolar potentials is justified by the fact that these are long range interactions that can exert their action on a length scale much larger than the typical dimensions of biomolecules. In this framework other interactions, i.e. Van der Waals interactions, have a very short range and they exert their action on length scale comparable with biomolecules dimensions. Nevertheless the short range screened Coulombic potential has been investigated as its range distance depends on the free ions concentration in the diffusive medium, which is an accessible experimental parameter. In what follows the diffusion of tracers in presence of interacting obstacles is studied for some cases corresponding to the different frameworks discussed in Sections “Modification of the microscopic freeflight time distribution”, “Modification of the rescaled freeflight time distribution”.
Case of modification of the microscopic freeflight time distribution
where q is the electric elementary charge and ε _{ water }≃80 is the relative electric permittivity of water.
Assuming that the friction coefficient is given by Stokes’ law (15), the obtained Γ value corresponds to η≃1.5×10^{−4} η _{ water }, where η _{ water } is the viscosity of water at temperature T=300 K.
We can conclude that the selfdiffusion coefficient of tracers is mainly affected by the value of the friction coefficient. In the range of cases we have studied, the presence of interactions among obstacles affects only slightly the diffusion behavior of tracers, as it can be seen by comparing with the case \(\bar {\mathcal {C}}_{Coul}=0\). This effect can be interpreted as a sort of “effective dynamical excluded volume” due to the presence of the obstacles; when the friction forces are weakened, the average speed both of the obstacles and the tracers increases and as a consequence the average freeflight time of tracers diminishes.
where λ _{ D } is the characteristic screening length scale, also called Debye length.
Case of modification of the rescaled freeflight time distribution
As discussed in Section “Modification of the rescaled freeflight time distribution”, the proposed approach corresponds to the case where the characteristic timescale τ of Brownian collisions is much smaller than the transition time T _{ d }. This corresponds to intermolecular distances d of the obstacles that are much larger than \(\sqrt {\frac {m_{A} k T}{{\gamma _{A}^{2}}}}\,\).
Conclusions
The main aim of this paper is to give an analytical estimation of the selfdiffusion coefficient of passive tracers as a function of the concentration and strength of mutual interaction of obstacles. We considered a very simple model of passive tracers and interacting obstacles diffusing in a low concentration limit. The diffusion law was assumed to be Brownian both for the tracers and the obstacles. Nevertheless, it would certainly be interesting in further studies to consider also other diffusive laws, in order to refine the model for crowded systems [20]. The CTRW framework is well suited for this, as it has been discussed in Section “Methods: continuous time random walk formalism”.
We found that the value of the Brownian selfdiffusion coefficient of passive tracers is markedly affected by the randomly moving obstacles. The effects related to the presence and the strength of interactions among obstacles is in general less important than the “effective dynamical excluded volume” related to the friction constant. We stress that this result strongly depends on our estimation of the freeflight time T _{ d } of passive tracers, which is quite crude and seems to be the main aspect to be refined in our model in order to obtain more accurate results. An attempt to modify the estimation of T _{ d } is suggested in this article, resulting in the so called rescaled freeflight time distribution; in this case, the effect of friction is neglected and the slowing down of the passive tracers diffusion is due to only to the concentration of the obstacles and the strength of their mutual interactions. Nevertheless, this model has not yet a clear correspondence to real biological models.
Although we have adopted strong approximations and simplifications with respect to a realistic biological case of crowding, this work represents a first step in the analytic study of the value of the diffusion coefficient of passive tracers in the presence of interacting obstacles, and this fact can have relevant prospective consequences for applications to biology. For instance, the description of the complex network of biochemical reactions taking place in living cells could be markedly affected by the activation of longrange intermolecular interactions of the kind discussed in Ref. [15]. In particular, if we imagine a cytoplasm crowded by biomolecules interacting at a long distance, then molecules that would be driven to their targets only by diffusion could be considerably slowed down.
Appendix
In this section, we compute the probability distribution P(r,t) for the walker to be at location r, at time t, following [17] and generalising the result to the threedimensional case.
Jump model
In the Jump Model, particles wait at a particular location before moving instantaneously to the next one, the displacement being chosen according to the probability density Λ(r), the waiting time before the jump being r/v _{0} (because of the δfunction in the expression of ψ(r,t)).
where Δ _{ k } is the Laplacian (\(\Delta _{\textbf {k}} = \partial ^{2}/\partial {k_{x}^{2}} + \partial ^{2}/\partial {k_{y}^{2}} + \partial ^{2}/\partial {k_{z}^{2}}\)) and ∇_{ k } is the gradient (∇_{ k }=(∂/∂ k _{ x },∂/∂ k _{ y },∂/∂ k _{ z })).
where ϕ(s) is the Laplace transform of ϕ(t).
Velocity model
where ϕ(s) is the Laplace transform of ϕ(t).
Declarations
Acknowledgments
The authors wish to thank F. Piazza and R. Lima for useful comments and suggestions. This work was supported by the Seventh Framework Programme for Research of the European Commission under FETOpen grant TOPDRIM (Grant No. FP7ICT318121).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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