On the importance of the diffusivity gradient term in Brownian dynamics simulations

Abstract

In a recent study, Garner et al. investigated diffusion in the cytoplasm of fission yeasts, revealing vast heterogeneity in intracellular viscosity. Their conclusion was based on a combination of single-particle-tracking experiments and Brownian dynamics simulations. However, in their simulations, the diffusivity gradient term has been neglected—an assumption common in some biophysical applications but unjustified in this particular case due to spatial variations in diffusivity. Here, we aim to comment on the importance of the diffusivity gradient term and the physical consequences of not including it. We also demonstrate that omitting this term likely leads to overestimating fission yeast intracellular viscosity variance and underestimating its mean. Additionally, we propose modifications to the simulations to incorporate the gradient term.

Main text

In a recent publication in the Biophysical Journal, Garner et al. investigated the diffusion of genetically encoded multimeric nanoparticles (GEMs) in the cytoplasm of fission yeasts Schizosaccharomyces pombe employing single-particle tracking (1). Their study revealed a surprisingly high variance of GEMs’ diffusion coefficients, both at the intracellular level and across different cells. Subsequently, the researchers conducted “Doppelgänger” Brownian dynamics (BD) (2) simulations of point particles within systems segmented into sectors characterized by different, randomly assigned viscosity. Through the lens of these simulations, Garner et al. assessed the extent of viscosity heterogeneity required to replicate the observed results.

Although acknowledging the ingenuity of Garner et al.’s overall reasoning, our motivation to write this commentary stems from the neglect of a certain term in the equation used to propagate the simulations presented in the discussed paper. Specifically, the BD propagation scheme employed by Garner et al. did not include the term with the gradient of the diffusion coefficient (3). This assumption should not have been made because of the spatial variation of diffusion coefficient attributed to viscosity heterogeneity.

The purpose of our paper is three-fold. Firstly, we discuss the diffusivity gradient term, outlining its importance and the physical consequences of neglecting it. Secondly, we hypothesize about the impact of its omission on the conclusions regarding heterogeneity in intracellular viscosity presented by Garner et al. Finally, we propose modifications in BD simulations to account for the gradient term in future studies.

Diffusivity gradient term

To prove our point, a detailed examination of the BD protocol applied by Garner et al. is essential. In their BD simulations, the authors partitioned a simulated system into a grid of N subsystems of different dynamic viscosity ${\eta }_i$, randomly selected from a log-normal distribution. The mean and standard deviation of the distribution were adjusted to match the experimental mean-squared displacement (MSD) statistics.

The authors employed a first-order forward Euler scheme for BD propagation:

$$\mathbf{r}\left(t+\Delta t\right)=\mathbf{r}\left(t\right)+{\left[2k_{\mathrm{B}}T\Delta t/\gamma \left(\mathbf{r}\right)\right]}^{1/2}\mathbf{X}\text{,}$$ (1)

where $\mathbf{r}$ is position, t is time, $\Delta t$ is timestep, $k_{\mathrm{B}}$ is Boltzmann constant, T is temperature, γ is friction coefficient, and $\mathbf{X}$ is standard normal random variable. For spherical particles, the Stokes-Sutherland-Einstein equation yields the following:

$$\gamma \left(\mathbf{r}\right)=6\pi \eta \left(\mathbf{r}\right)R\text{,}$$ (2)

with R being a particle’s hydrodynamic radius. The term involving the gradient of diffusion coefficient ($D=k_{\mathrm{B}}T/\gamma$) was neglected, a departure from the originally derived BD propagation scheme by Ermak and McCammon (4):

$$\mathbf{r}\left(t+\Delta t\right)=\mathbf{r}\left(t\right)+\nabla D\left(\mathbf{r}\right)\Delta t+{\left[2D\left(\mathbf{r}\right)\Delta t\right]}^{1/2}\mathbf{X}\text{.}$$ (3)

Neglecting the gradient term is common in biophysical applications, where diffusion coefficients (or diffusion matrices in many-body systems) are often assumed constant or a sourceless tensor field, such as the Oseen or Rotne-Prager-Yamakawa tensor (5), for which the gradient vanishes ($\nabla \mathbf{D}=0$). However, this assumption is not universally valid, and in the paper by Garner et al., it should not have been made.

To underscore the importance of the gradient term, consider a system divided into two equal-size compartments with different viscosities ${\eta }_1<{\eta }_2$, respectively, and with a reflective boundary separating it from the environment. Without the gradient term, as the particle performing Brownian motion moves ${\eta }_2/{\eta }_1$ times quicker in the first compartment, it will spend ${\eta }_2/{\eta }_1$ times more time in the second one. The simulated particle’s position distribution becomes biased, instead of approaching the uniform distribution in the limit of infinite time, in accordance with the Boltzmann formula.

We can reverse-engineer what bias potential $\mathcal{V}\left(\mathbf{r}\right)$ has to be exerted on the system in order to equalize Boltzmann weights for both subsystems. Without the gradient term, the propagation leads to ${\mathcal{P}}_1\propto {\eta }_1$ and ${\mathcal{P}}_2\propto {\eta }_2$, so, to cancel out this dependency, the bias potential has to fulfill the following condition:

$$\exp \left[-\mathcal{V}\left(\mathbf{r}\right)/k_{\mathrm{B}}T\right]\propto 1/\eta \left(\mathbf{r}\right)\text{.}$$ (4)

Solving the above equation for $\mathcal{V}$ gives

$$\mathcal{V}\left(\mathbf{r}\right)=k_{\mathrm{B}}T\ln \eta \left(\mathbf{r}\right)\text{.}$$ (5)

The external force, such as the analyzed bias, is introduced into the BD simulation scheme multiplied by timestep and the mobility, i.e., $\mathit{D}/k_{\mathrm{B}}T$ (4). After computing the derivative of Eq. (5), the deterministic term reads

$$D\left(\mathbf{r}\right)\mathbf{F}\left(\mathbf{r}\right)\Delta t/k_{\mathrm{B}}T=\nabla D\left(\mathbf{r}\right)\Delta t\text{.}$$ (6)

This bias potential derived with our reverse-engineering reasoning reproduces the gradient term from Ermak and McCammon’s paper (4), which was not included by Garner et al. (1).

Effect on Garner et al.’s conclusions

To evaluate the impact of neglecting the gradient term on the conclusions drawn by Garner et al. (1), we briefly review their key findings. The authors demonstrated that the experimentally observed variance in the distribution of diffusion coefficients exceeds that of BD simulations assuming spatially uniform viscosity. They addressed this disparity by partitioning the BD system into N segments characterized by different viscosity ${\eta }_i$, sampled from a log-normal distribution to match the intracellular variance. Their results suggested a variability of over 100-fold in the effective cytoplasmic viscosity within cells.

To gain insight into the possible effect of not including the gradient term on these results, we begin from the 2D solution of the diffusion equation originating from a point source (placed at the center of the coordinate system for convenience), which reads

$$\mathcal{P}\left(x,y;D,t\right)={\left(4\pi Dt\right)}^{-1}\exp \left[-\left(x^2+y^2\right)/4Dt\right]\text{.}$$ (7)

The MSD can be computed from this relation as

$$\text{MSD}={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left(x^2+y^2\right)\mathcal{P}\left(x,y\right)dxdy\text{,}$$ (8)

leading to the result: $D=\text{MSD}/4t$, a formula utilized by Garner et al. to extract the diffusion coefficient from the experimentally measured MSD. Note that Eq. (7) can be transformed to a probability density distribution of squared displacement:

$$\mathcal{P}\left(\text{SD};D,t\right)={\left(4Dt\right)}^{-1}\exp \left(-\text{SD}/4Dt\right)\text{,}$$ (9)

which takes the form of an exponential distribution.

The intrinsic variance of diffusion coefficient can be calculated as the variance of squared displacement:

$${\sigma }_{\text{SD}}^2=⟨{\text{SD}}^2⟩-{⟨\text{SD}⟩}^2={\left(4Dt\right)}^2\text{,}$$ (10)

which yields the following expression for the standard deviation of the diffusion coefficient:

$${\sigma }_D=D\text{,}$$ (11)

in accordance with the known properties of exponential distribution.

Without diving into numerical details, we can still gauge the effect of not including the gradient term by careful analysis of the equation introduced above. Let us assume that the tracks are short, and the probability of observing a single track that moves through a few different sectors is low. Then, the probability density distribution of SD is a weighted average of Eq. (9):

$$\mathcal{P}\left(\text{SD};\left\{c_i\right\},\left\{D_i\right\},t\right)=\sum _{i=1}^N{c}_i\mathcal{P}\left(\text{SD};D_i,t\right)\text{,}$$ (12)

where the weights $c_i$ correspond to the fraction of time spent in the respective segments. The mean of the probability density distribution expressed with Eq. (12) reads as follows:

$$⟨\text{SD}⟩=4{⟨D_i⟩}_{c_i}t\text{,}$$ (13)

and the standard deviation reads

$${\sigma }_{\text{SD}}=4{\left(2{⟨D_i^2⟩}_{c_i}-{⟨D_i⟩}_{c_i}^2\right)}^{1/2}t\text{.}$$ (14)

From the previous section, it is established that the true coefficients $c_j$, assuming uniformly sized segments, are equal to $1/N$ and independent of ${\eta }_j$. However, not including the gradient term (Eq. (1)) gives $c_j$ of the following form:

$$c_j={\eta }_j/\sum _k^N{\eta }_k=D_j^{-1}/\sum _k^N{D}_k^{-1}\text{.}$$ (15)

The question we aim to answer is how the diffusion coefficient mean and variance in heterogeneous system would change upon inclusion of the gradient term in the propagation scheme.

Neglecting the gradient term, the weights $c_j$ skew the weighted averages (Eqs. (13) and (14)) in the direction of low $D_i$ corresponding to high viscosities, which shows that the mean and variance of D while propagating BD with the gradient term is necessarily larger than when neglecting it. It is thus possible that Garner et al. needed to increase the width of their distribution of viscosities too much to match their experimental dynamics. Moreover, our analysis questions the choice of the distribution’s mean value as well.

Our analysis suggests that the width of the intracellular viscosity distribution uncovered by Garner et al. might be overestimated, and its mean is likely underestimated. However, we humbly admit that our analysis was to a large extent oversimplified, e.g., we do not account for spatial organization of the domains or for the tracks engaging more than one sector. We encourage Garner et al. to explore the extent of this effect further.

Modification of the simulations

One question that naturally arises is how to incorporate the gradient term into the simulations. The most straightforward approach involves simply changing the propagation scheme from Eq. (1) to Eq. (3). This term does not alter the dynamics within segments, where viscosity remains constant. However, at the boundaries between segments, particles are effectively drawn toward regions of lower viscosity. As a result, transitions to thicker segments become less probable, and the correct statistics are recovered.

Caution is needed while implementing this change because to this point, the viscosity is defined as a discontinuous function of position. In order to correct for that, the viscosity should be interpolated in the regions neighboring the boundaries. The width of these intermediate regions needs to be substantially smaller than the mean Brownian displacement in the region of smaller viscosity, for the particle to be unable to leap through them without experiencing the gradient term. To ensure that the derivative term is correctly implemented, we recommend verifying if the spatial distribution of positions approaches uniform distribution in $t\to \infty$ limit.

Conclusions

In conclusion, our examination of the BD protocol employed by Garner et al. has uncovered neglect of the diffusion coefficient gradient term. Not including this term, although common in certain biophysical applications, is not justified in scenarios with spatially varying diffusion coefficients because it results in unphysical heterogeneity in position distribution, inconsistent with the Boltzmann distribution. Furthermore, our simple analysis of diffusion in systems with heterogeneous viscosity reveals the consequences of this omission for Garner et al.’s conclusions, namely the likely overestimation of intracellular viscosity variance and underestimation of its mean.

Our findings also address the work of Venkatesh et al. (6), who investigated the effects of inhomogeneous viscosity on particle mobility and concentration and introduced the concept of diffusive lensing to describe particle aggregation in regions of higher viscosity. This interpretation relies on the same neglect we discuss in our paper. Including the gradient term guarantees that the position distribution tends toward the Boltzmann distribution, which is independent of viscosity and depends solely on potential energy.

Editor: Guy Genin.