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Superstatistical modelling of protein diffusion dynamics in bacteria

A recent experiment [A. A. Sadoon and Y. Wang, Phys. Rev. E 98, 042411 (2018)] has revealed that nucleoid associated proteins (i.e., DNA-binding proteins) exhibit highly heterogeneous diffusion processes in bacteria, where not only the diffusion constant but also the anomalous diffusion exponent fluctuates for the various proteins. The distribution of displacements of such proteins is observed to take a q-Gaussian form, which decays as a power law. Here, a statistical model is developed for the diffusive motion of the proteins within the bacterium, based on a superstatistics with two variables. This model hierarchically takes into account the joint fluctuations of both the anomalous diffusion exponents and the diffusion constants. A fractional Brownian motion is discussed as a possible local model. Good agreement with the experimental data is obtained.


Introduction
Diffusion phenomena in living bacteria pose challenging scientific problems (see, for example, Refs. [1][2][3][4][5]). There is a rapidly growing development of the experimental techniques of single-particle tracking (see Ref. [6] and references therein) which has made new interesting experimental results available. Recently, highly heterogeneous diffusion processes have been observed in various experiments (see e.g. Ref. [7] and references therein), for example in the experiment of Ref. [8] (see also Ref. [9] for a recent update), for the dynamics of histonelike nucleoid-structuring proteins in living Escherichia coli bacteria. Such nucleoid associated proteins interact with DNA as well as with themselves and they are uniformly distributed over the bacterium. In the experiment, to measure the diffusion properties, the trajectories of these proteins, fused to fluorescent proteins, have been analyzed at individual level. An interesting observation of Ref. [8] is that the displacement, , x  is not Gaussian distributed but obeys a q -Gaussian distribution [10] (sometimes also called a Pearson-type VII distribution [11]): where w is a positive quantity having the dimension of space and m is a positive exponent. This density asymptotically decays with a power law, This behavior is a priori unexpected, since the power-law implies the non-existence of a characteristic displacement and, accordingly, it is in marked contrast to the results obtained in other experimental studies, for example, in Refs. [12][13][14], where the distribution of displacements of mRNA or chromosomal locus in living bacteria was 3 found to be of exponential form, with a characteristic displacement. Thus, the result of Ref. [8] sheds new interesting aspects onto the complexity of protein diffusion in living bacteria.
The analysis of Ref. [8] has shown that there is actually a distribution of different anomalous diffusion exponents, as well as a distribution of different diffusion constants in the bacterium. In this paper we will provide a theoretical framework for analyzing these distributions.

Experimental observations
Let us write for the mean square displacement of a given protein where  D is the diffusion constant,  is the (anomalous) diffusion exponent, and t is elapsed time. (Here and hereafter, the notations we use are slightly different from those in Ref. [8].) Normal diffusion implies , 1   whereas the case with 1   ( 0  ) corresponds to anomalous diffusion [15][16][17].
A fundamental experimental observation of Ref. [8] is that not only the diffusion constants but also the diffusion exponents fluctuate in a wide range. The distribution of the diffusion constants is observed to asymptotically follow a power law 1 ) ( It may be worth pointing out that the power-law nature in Eq. (3) is nontrivial, since it is apparently different from the exponential law reported, for example, in Ref. [13] (see also Refs. [18,19] for an entropic approach to this law).
The mean square displacement in an ensemble average, i.e., an average of square displacement over all of the individual trajectories, has also been obtained in the measurements of Ref. [8], where both an average diffusion constant and an average diffusion exponent are determined. Then, the bacteria have been classified into three groups based on their cell age (or, equivalently cell length). As can be seen in Figs. 6(c) and 6(d) in Ref. [8], it has been observed for such groups that, in terms of the cell age, the average diffusion constant increases significantly, whereas the average diffusion exponent is approximately constant (it increases only slightly).

Two-variable superstatistical treatment
Generally, in nonequilibrium statistical physics the diffusion constant is proportional to temperature through the Einstein relation [20], hence a theory of fluctuating diffusion 5 constants is mathematically equivalent to a theory of fluctuating temperatures. This leads us naturally to the concept of superstatistics [21,22]. Superstatistics is a "statistics of statistics" with largely separated time scales: a prototype example is a Brownian motion in a fluid environment in a variety of nonequilibrium stationary states [23]. The marginal distribution of the Brownian particle is written as a superposition of the statistics describing the local Brownian motion on a short time scale (where inverse temperature is locally fixed) with respect to the statistics associated with slowly fluctuating inverse temperatures on a long time scale. The superstatistical idea has been widely used for a variety of complex systems/phenomena organized by different dynamics with such a separation of time scales (see Ref. [24] for a review and Refs. [25][26][27][28][29][30] for very recent developments). Among others, the discussions in Refs. [7,31] are suitable for biological application: in the former, a fluctuating quantity is the diffusion constant, whereas it is the diffusion exponent in the latter.
The basic idea of this paper is to develop a two variable-superstatistical formalism which is then applied to model the stochastic motion of histonelike nucleoid-structuring proteins in living Escherichia coli bacteria. In our approach, the time scale of variation of the fluctuations of both the diffusion exponent and the inverse temperature is much larger than that of the dynamics of the protein in local areas of the bacterium, leading to associated statistics with two largely-separated time scales. In fact, the superstatistical probability densities can simply be associated with the different types of proteins that diffuse in the bacterium, thus there is naturally an ensemble of different diffusion constants and different scaling exponents of anomalous diffusion. We describe the statistical property of the protein over the bacterium as a superposition of these different statistics. For a given protein in given local areas, fractional Brownian motion [32] (a 6 simple Gaussian stochastic process where the mean square displacement exhibits anomalous diffusion) is used as a simple local stochastic process, see also Ref. [7] and references therein, but our approach can be easily generalized to other local stochastic models. We show that the present theory gives rise to q -Gaussian (Pearson-type) distributions, in agreement with the experimental measurements. We also propose a Generally, it is well-known that the diffusion constant for ordinary Brownian motion is proportional to temperature, the proportionality constant being the mobility of the For the protein (or ensemble of proteins) in a given local block, we denote the probability of finding the protein in the interval The protein moves from one block to another on a long time scale.
Therefore, for the entire bacterium let us introduce a superstatistical ensemble of all proteins by defining the following integrated probability distribution: Equation (5)  So far Eq. (5) is formal, without having determined the distributions appearing there. 8 In the following, we discuss distributions that are consistent with the experimental data.
is assumed and the prime denotes, throughout the present work, differentiation with respect to .  In contrast to In the following we use the fact that the above experimentally observed power-law distribution in bacteria is only describing the behavior for small values of the inverse temperature  (i.e. large values of temperature T ) . We are free to assume suitable behavior for larger values of  to get agreement with q -Gaussians of the displacement distributions, by integrating over all .
 This is the standard formalism of superstatistics for one variable as described in Ref.
[23], assuming a 2  -distributed .  Therefore, let us suppose that Euler gamma function. The above functional form exhibits power-law behavior for small ,  as required, but also provides a cut-off for large .  We also mention that the 9 experimental data is seen to be consistent with Eq. (6), see Fig. 3 as well as Appendix B. is assumed to be small.

A possible model
as supported e.g. by the data in Ref. [8] and examined for the present case in Appendix D, we apply the approach of fractional Brownian motion [32] as a possible stochastic process. By this we mean that the probability density is given as follows [17]: denotes the diffusion constant. It is given by the relation s and c being a characteristic time and a positive constant, respectively, and it corresponds to  D in Eq. (2) (see Appendix A for this relation).
Substituting Eqs. (6)-(8) into Eq. (5), we obtain the following distribution: where the symbol   denotes the average with respect to the conditional distribution provided that, following the formalism in Ref.
[23], the integration over  has been performed, and the quantity, has been assumed to be positive [which is indeed confirmed in the case of Eq. (15) below, see Appendix F].
Some properties of the experimental data are discussed in Appendix E. Based on this, in the region of the local blocks with constant diffusion exponent, we can take the conditional distribution in Eq. (7) in good approximation as follows: From this, as well as Eqs. (9) and (10), we immediately derive the following distribution: Clearly, Eq. (11) has the form of a q -Gaussian distribution. In Fig. 4, we present the plot of in Eq. (11). The experimental data is nicely described by the q -Gaussian distribution in Eq. (11). Upon renaming, (1), we therefore have the following relations: Equation (12)   To examine Eq. (9) in this situation, all kinds of conditional distributions can be considered. In the following we wish to discuss a particular example form of the conditional distribution ). | (   g As discussed in Appendix F, this is given by where ) ( a is a positive quantity depending weakly on  and Finally, in Fig. 5, we present the plot of the distribution of Eq. (9) using Eqs. (14) and (15). This long-term distribution is seen to decay somewhat slower as compared to Eq. (11). Further experimental data of protein anomalous diffusion dynamics can help to build 13 the optimum model for a given experiment, possibly checking for universal and nonuniversal properties, i.e. properties that are observed for all bacteria and other properties that are very specific to a given specific experiment.

Conclusion
We suggests that this fluctuation comes from displacement of the protein rather than the characteristic time being required for displacement, since such a characteristic time is constant in the random walk picture for normal diffusion [37]. The situation may be same for  D due to the same power-law nature of diffusivity. It has been found, for the mean square displacement in the ensemble average, that the numerical value of the diffusion constant in the case of normal diffusion observed for small elapsed time is three times larger than that in the case of anomalous diffusion observed for large elapsed time [8]. Therefore, for a given individual trajectory, these observations seem to allow us to evaluate ,

APPENDIX B
The left-bottom inset in Fig. 2(d)  suppose that such a curved line is described by the following inverse gamma distribution A is a positive constant having the dimension of the 16 diffusion constant,  and  are lower and upper bounds on , D respectively, see Fig.   3. Clearly, this distribution decays as a power law for large . D Here, we employ the assumption that the range of  in Eq. (6) is unbounded, although the values of  and  in practice (i.e. in true experiments) are of course small and large, respectively, but still finite.
From the experimental data in Fig. 2(d) in Ref. [8], the situation seems to be similar As this curved line, we shall take the inverse gamma distribution given by we have the conditional distribution in Eq. (7), and the weak correlation is described in this way.
From Eqs. (6) and (7), the marginal distribution ) ( g is immediately calculated to be we examine here if the approach of fractional Brownian motion [32,38] is appropriate for the typical local dynamics in a bacterium. A process of fractional . In our present discussion, the following three features of the process of fractional Brownian motion are relevant. Firstly, the velocity autocorrelation function in the process becomes negative (positive) if 2 [17]. This is consistent with the experimental result presented in Fig. 5(a) in Ref. [8], where it is clearly seen that the velocity autocorrelation function of the proteins becomes negative for small elapsed time. It should be also noticed that the case of 1 0    has been predominantly observed. Secondly, the diffusion processes are assumed to be ergodic, since the protein can diffuse almost anywhere, in the sense [17,40] that the mean square displacement obtained through the position autocorrelation function is equivalent to the one derived based on the probability distribution. Lastly, the process does not exhibit any aging phenomenon, since the sequence of the increments of ) (t B H is stationary [17].
Regarding this point, we should say that for the experimental data [8] there is a dependence on cell age, but this can be incorporated in the choice of the effective temperature relevant for each cell age group.
Taking into account all of these considerations, we see that the model of local fractional Brownian motion is a good one for the local dynamics modelling in bacteria and other small complex systems. It simply needs to be amended by the detailed properties of the 19 diffusion constant distribution and diffusion exponent distribution.

APPENDIX E
We discuss some further experimental features of the data in Ref. [8]. Based on the experimental data presented in Figs. 6(a), 6(c), and 6(d) in Ref. [8] and Eq. it seems natural to consider, for such a large time, that the protein diffuses over a region of a few local blocks in the bacterium.
This implies that the fluctuations of ,  rather than those of ,  dominantly contribute to the protein dynamics due to its power-law nature. In the latter, we employ the average value, ,  of the diffusion exponent for the bacteria in each group based on the experimental data in Fig. 6(d) in Ref. [8].

APPENDIX F
Here, we present a particular form of the distribution   (14) and (15) takes on a maximum value and a local minimum value, respectively, at     and : where the quantity inside the square root in Eqs. (F3) and (F4) is assumed to be positive (which is confirmed in the present case, see below). 22 In Fig. 2 [8], see the histogram in Fig. 2.
The experimental data imply that the quantity

ACKNOWLEDGEMENT
The present research has been designed jointly by Y.I. and C.B. and it was completed while Y.I. stayed at the Institut für Computerphysik, Universität Stuttgart. Y.I. would like to thank the Institut für Computerphysik for their warm hospitality.       Fig. 4(a) in Ref. [8]. The three lines show the q -Gaussian distributions of Eq. (11) ) are considered here.