Diffusion in crowded and complex systems such as biological cells is usually very heterogeneous, and anomalous behavior—where the mean square displacement of tracers varies non linearly with time—is envisaged [1–3]. Over the last few years a new class of diffusive processes has been reported, where the mean square displacement is found to grow linearly in time like in standard, Brownian diffusion, but with a corresponding probability density function (PDF) which is strongly non-Gaussian [4–16]. This behavior, termed Brownian yet non-Gaussian diffusion [6, 8], occurs quite robustly in a wide range of systems, including beads diffusing on lipid tubes [6] or in networks [6, 7], the motion of tracers in colloidal, polymeric or active suspensions [4, 17–19] and in biological cells [12, 20, 21], as well as the motion of individuals in heterogeneous populations such as nematodes [5]. Similar effects on the PDF are also observed in the anomalous diffusion [22] of labeled messenger RNA molecules in living E.coli and S.cervisiae cells. In the majority of cases, at larger time the form of the PDF crosses over to the normal, Gaussian one. Therefore, such change cannot be simply due to the heterogeneity of the tracers, unless some of their properties vary with time. More plausibly, the anomalous-to-Gaussian transition might be induced by temporal fluctuations of the diffusion coefficient, due to rearrangements of properties of tracers or of the surrounding medium. To mimic such behaviors, models in which the diffusion varies with time by obeying a stochastic equation have been introduced and solved both analytically and numerically. These models are referred in the literature as “diffusing diffusivity models” [23–32], and it has been shown that for short times they are intimately related to the idea of superstatistics [33]. In the latter approach, an ensemble of particles is assumed to be characterized by different diffusion coefficients and it is then described as a mixture of Gaussian PDFs, weighted by the distribution of the diffusivities. As a result, the ensemble dynamics is still Brownian, yet the PDF of particle displacements corresponds to a Gaussian mixture and it is thus not Gaussian anymore.

Although diffusing diffusivity models qualitatively reproduce the experimental observations, they work at a mesoscopic scale and without a visible connection to the underlying molecular processes. It is therefore becoming increasingly relevant to find strategies that bridge the gap between the paradigm of diffusing diffusivity and the microscopic realm, in order to fully understand this form of anomalous diffusion. In this paper we show how the diffusion of polymers during a polymerization process offers one possible mechanism to realize this connection¹. It is well known from polymer theory [36] that the motion of the center of mass of a linear chain is Brownian, but with a diffusivity constant which is inversely proportional to N^α, where N is the number of monomers and α an exponent ranging from 1/2 (Rouse model) to 2 (reptation model). During an equilibrated polymerization processes the number N fluctuates in time and its statistics can be obtained through the exact solution of its stationary master equation. By using a continuous approximation for this temporally homogeneous birth-death Markov process [37], it emerges that in the limit of large systems such process converges to an Ornstein-Uhlenbeck, as it is assumed in most of the diffusing diffusivity models [24]. The time scale of the Ornstein-Uhlenbeck process is linearly proportional to the volume of the system and this guarantees that the non-Gaussian behavior can be accessible experimentally by tuning such parameter.

Polymers are made of relatively simple subunits (monomers) assembled with one another through different mechanisms and geometries. The result is a macromolecule which may contain from a few tens (in the case oligomers), to several thousand monomer units [38], or even millions as in the case of DNA and RNA molecules. From a biological point of view, the polymerization process occurs regularly either within or outside the cell [39]. In particular, cells might trigger polymerization by several mechanisms such as the de novo nucleation of new filaments, the uncapping of existing barbed ends (actin) and rescuing a depolymerizing filament (commonly observed for microtubules).

In order to guarantee the existence of equilibrium conditions, here we consider a polymerization process occurring in a closed volume with a fixed total number of monomers N_t. For sake of simplicity, in what follows we suppose that one filament only can nucleate and that subunits may bind reversibly onto both ends of the chain. At each end, the addition and deletion of monomers can be represented as [40]

where A_N is the filament with N subunits, and k₊, k₋ are the rate constants for association and dissociation, respectively. Hence,

where M(t) = c(t)V is the number of monomeric subunits, c its concentration and V the system volume. The probability of a filament with n monomers at time t given n₀ units at time t₀, P_N(n, t|n₀, t₀) satisfies the (forward) master equation of a temporally homogeneous birth-death Markov process [37]:

with stepping functions

and c(n) = (N_t − n)/V. Through these choices, we are assuming with certainty the existence in solution of a filament with at least one monomer. The factor 2 in W₊ models a linear polymer which grows at both ends without developing branching; W₋ is instead concerned with the possible bonds which may break down. Equilibrium is reached under detailed balance W₋(n) = W₊(n) (3 ≤ n ≤ N_t), corresponding to a polymer composed by

monomers, and to a number

of single monomers in solution. We remark that the rate constants k₊, k₋ are specific to the polymerization chemical reactions. Given a certain kind of polymer, the average polymer size and the average number of single monomers in solution are thus controlled by the total number of subunits N_t and by the volume of the system V, which are quantities easily controlled in experiments. In the following analysis, we find it convenient to replace the volume with the fraction 0 < λ < 1 of N_t that compose the polymer at equilibrium; clearly, V = (1 − λ) N_tk₊/k₋.

As we prove in the Appendix, for any given N_t and independently from n₀, the stationary solution PN(n)≡limt→∞PN(n,t|n0,t0) reads

with a normalization factor

Γ(·, ·) being the upper incomplete gamma function [41],

and Γ(·) the Euler gamma function. We may observe that with (1 − λ)N_t → 0 the two Gamma functions in the normalization factor become equal and simplify to 1; in this limit, probabilities for small n are suppressed. Indeed, in section 4 we show that P_N(n) becomes close to a Gaussian for large λ and N_t. In view of the inverse power-law relation with the diffusion coefficient of the center of mass, it is however the behavior for small n which affects the probability of large diffusivities, triggering in turn strong deviations from ordinary diffusion which are described in the following Section.

From polymer physics we know that the center of mass R_G of a macromolecules with N subunits diffuses with a coefficient D(N)=D0/Nα, D₀ being a diffusion coefficient specific of the considered subunit. This means

with B(t) a (three-dimensional) Wiener process (Brownian motion). Reference values for the exponent α are:

In view of the previous analysis, we understand that polymerization confers a random character to R_G, providing a clear microscopic origin to the “diffusing diffusivity” process we are going to detail next.

From Equation (7) we readily obtain the stationary distribution for the diffusion coefficient of the polymer's center of mass,

and its first moment

Imagine now to perform a particle-tracking experiment at constant N_t and V and to monitor the position of R_G in stationary conditions. At a given initial instant the polymer possesses a size n, and thus a diffusion coefficient Dn=D0/nα with probability given by Equation (12). For time smaller than the characteristic decay τ of the autocorrelation of the process N(t), the experimental PDF amounts then to a Gaussian mixture (also called “superstatistics”) [6, 23, 33] weighted by Equation (12). In addition, its second moment grows linearly with time as in the ordinary Brownian motion. Such a phenomenon of “Brownian yet non Gaussian diffusion” [6, 8] has been recently modeled at a mesoscopic scale in terms of diffusing diffusivity models [23–32]. It is only at time larger than τ that ordinary (Gaussian) Brownian motion is recovered, with a diffusion coefficient D_av. Before giving an estimate of τ for our model (see next section), we study the early time non-Gaussianity in the full phase diagram [N_t, λ], together with its dependence on α.

The non-Gaussian behavior distinctive of R_G(t) at time 0 ≤ t ≪ τ can be properly characterized by referring to one of its Cartesian coordinates, say x. The PDF of the x-displacements takes the form

In Figure 1 we plot Equation (13) for α = 1 and different values of λ and N_t. At first sight, non-Gaussianity increases with decreasing N_t and λ; below we however show that the behavior is not monotonic. To measure deviations from Gaussianity we consider the kurtosis of p_X(x, t),

(κ = 3 for any Gaussian variable). In our case it is straightforward to see that

independently of D₀. Notice instead the strong dependence of κ from α; moreover, κ > 3 (positive excess kurtosis or leptokurtic PDF). In order to illustrate regions of more pronounced non-Gaussianity and to discuss their dependence on α in Figure 2 we draw the kurtosis level curves within a (λ, N_t)-phase diagram. Note that, for a given pair (N_t, λ), higher values of the exponent α give rise to larger kurtosis (compare Figures 2A,B).

As quoted, by looking at the plots in Figure 1 one may expect the kurtosis to steadily increase by decreasing λ and N_t. The structure of the phase diagram implies instead the existence of a maximum kurtosis, both at given λ and N_t. Indeed, for any horizontal or vertical line traced through the phase diagram (Figure 2) it is possible to find a family of kurtosis level curves each intersecting the line in two distinct points. Between each couple of intersection points the kurtosis first raises and then decreases, thus reaching a maximum value. This is highlighted in Figure 3. Albeit within a small portion of the phase space, the maximum kurtosis can be extremely high, as reported in Figure 4; for instance, k_max ≃ 40 corresponds to an average polymer size of order N_eq ≃ 350 with Nt≃104.

The stationary distribution in Equation (7) is exact, but it does not provide information about the decay time-scale τ of initial conditions for the process N(t). To get such an insight, we next workout a continuous approximation for the polymerization process. In the gedankenexperiment reported above, τ is the persistence time scale of the randomly chosen initial diffusion coefficient for R_G, corresponding in turn to the typical duration of the leptokurtic PDF for the diffusion of the center of mass.

We start by noticing that around equilibrium, for N_t ≫ 1 and N_eq ≫ M_eq (large λ), N(t) can be approximated as a continuous Markov process with Langevin equation [37]

where B(t) is a Wiener process (Brownian motion). Taking further advantage of the large N_eq assumption, we then introduce the rescaled quantity N~≡N/Neq, obeying

to which we may apply the weak noise approximation. Indeed, one may straightforwardly prove [37] that for large N_eq Equation (18) is satisfied by the approximate solution

with n~(t) a deterministic process satisfying

and Y(t) the stochastic process defined by the Langevin equation

The solution of the deterministic process,

asymptotically tends to 1 with a characteristic decay time

Correspondingly, the long-time behavior of Y(t) is that of an Ornstein-Uhlenbeck process:

where ℕ(μ, σ²) is a Gaussian variable with mean μ and variance σ². Hence, the stationary solution of N~ is

For the polymer size N=N~Neq, this implies

We thus appreciate that, to be self consistent, the continuous approximation requires large values of N_t to blur out discreteness, and N_eq ≫ M_eq so that the negative support of the Gaussian PDF corresponds to a negligible probability. Figure 5 shows that when N_t and λ are both large the weak noise approximation of the stationary distribution P_N(n) is almost indistinguishable from the exact solution. On the other hand, decreasing either N_t or λ the approximation fails concomitantly with the fact that the Gaussian probability of negative n-values becomes significant. Depending on the specific cut in phase-space, the approximation may or may not work well in correspondence to the maximum kurtosis (compare red full lines in Figures 5A,B).

When applicable, the important result conveyed by the continuous, weak noise approximation is that through Equation (22) it establishes the time scale of the decay of the autocorrelation of N(t). It would be nice to give an explicit representation of τ in terms of the control parameters (λ, N_t); however, Equation (22) shows that it further depends on the dissociation rate constant k₋, which is specific to the chosen polymer. To get a qualitative insight, in Figure 2 we have added the line

representing the locus of points for which τ is equal to the inverse of k₋. Notice that the largest kurtosis level curve lay at the left of the line, a region which is also characterized by τ > 1/k₋. Hence, the farther left of the line the longer lasts the Brownian yet non-Gaussian diffusion stage.

We have been able to analytically characterize the stochastic motion of the center of mass of a fluctuating filament undergoing a simple polymerization process. Depending on experimentally accessible parameters such as the the total monomers in the solution N_t and the system volume V (equivalently, the fraction λ of total monomers composing the filament in equilibrium), the center of mass displays at early times a Brownian, yet non-Gaussian, diffusion. To our knowledge, this is one of the first example in which this anomalous behavior is directly linked to a microscopic prototype: the effect originates from the fluctuations of N (due to polymerization) and from the relation D(N)=D0/Nα which distinguishes many microscopic models of polymeric diffusion. By studying the kurtosis of the early-time displacement PDF along the x-coordinate we quantified deviations from Gaussian behavior in the phase diagram (λ, N_t), highlighting the dependence on the exponent α. Remarkably, the kurtosis is not monotonic and displays a maximum at either λ or N_t fixed. Finally, on the basis of a continuum (weak noise) approximation for the stochastic process N(t), we put forward an estimation for the time τ(λ, N_t) at which the anomalous behavior crosses over to ordinary Brownian motion. Since the weak noise approximation is not applicable in the whole (λ, N_t) phase diagram, and also in view of the non-monotonic behavior of the kurtosis, further studies approaching the determination of τ are welcome.

In parallel with the analytical results, we proposed a gedankenexperiment in which the anomalous behavior could be detected. As a further perspective, we may notice that if we shift the focus on the diffusion of a tagged monomer (in place of the center of mass of the polymer), in the early stage of the process a subdiffusive behavior coupled to non-Gaussianity is expected to be observed, with a crossover to a Brownian regime at the Rouse time [36]. This analysis is intended to be the subject of future work.

In conclusion, we believe that this work provides a valuable analytical backdrop to Brownian yet non-Gaussian diffusion, a fascinating phenomenon reported to occur in many physical systems. To fully understand this anomalous behavior, it is essential to ground it on a microscopic spring. This is the case for the presented model, but we are confident than others more will come along these lines.

The datasets generated for this study are available on request to the corresponding author.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.