Transient anomalous diffusion in run-and-tumble dynamics

We study the stochastic dynamics of a particle with two distinct motility states. Each one is characterized by two parameters: one represents the average speed and the other represents the persistence quantifying the tendency to maintain the current direction of motion. We consider a run-and-tumble process, which is a combination of an active fast motility mode (persistent motion) and a passive slow mode (diffusion). Assuming stochastic transitions between the two motility states, we derive an analytical expression for the time evolution of the mean square displacement. The interplay of the key parameters and the initial conditions as for instance the probability of initially starting in the run or tumble state leads to a variety of transient regimes of anomalous transport on different time scales before approaching the asymptotic diffusive dynamics. We estimate the crossover time to the long-term diffusive regime and prove that the asymptotic diffusion constant is independent of initially starting in the run or tumble state.

We study the stochastic dynamics of a particle with two distinct motility states. Each one is characterized by two parameters: one represents the average speed and the other represents the persistence quantifying the tendency to maintain the current direction of motion. We consider a runand-tumble process, which is a combination of an active fast motility mode (persistent motion) and a passive slow mode (diffusion). Assuming stochastic transitions between the two motility states, we derive an analytical expression for the time evolution of the mean square displacement. The interplay of the key parameters and the initial conditions as for instance the probability of initially starting in the run or tumble state leads to a variety of transient regimes of anomalous transport on different time scales before approaching the asymptotic diffusive dynamics. We estimate the crossover time to the long-term diffusive regime and prove that the asymptotic diffusion constant is independent of initially starting in the run or tumble state.

I. INTRODUCTION
Many transport processes in nature involve distinct motility states. Of particular interest is the run-andtumble process, which consists of alternating phases of fast active and slow passive motion. Prominent examples are bacterial species that swim when their flagella form a bundle and synchronize their rotation. The bundle is disrupted and swimming stops when some of the flagella stochastically change their rotational direction. In the absence of rotating bundle, the bacterium moves diffusively until it manages to re-form the bundle and actively move forward again [1,2]. The run-and-tumble dynamics is beneficial for bacteria as it allows them to react to the environmental changes by adjusting their average run time or speed [3], change their direction of motion, perform an efficient search [4][5][6][7], or optimize their navigation [8,9].
Another example is the motion of molecular motors along cytoskeletal filaments. When motor proteins bind to filaments, they perform a number of steps until they randomly unbind and experience diffusion in the crowded cytoplasm. While the efficiency of long-distance cargo delivery requires high motor processivity (i.e. the tendency to continue the motion along the filament), the slow diffusive mode during unbinding periods is also vital for cellular functions which depend on the localization of the reactants [10][11][12][13]. The processivity of the motors (and thus the unbinding probability) depends on the type of motor and filament [14,15] and the presence of particular proteins or binding domains in the surrounding medium [16][17][18]. On the other hand other factors, such as cell crowding, may affect the binding probability. Therefore, the switching probabilities between active run and tumble states are generally asymmetric. By ignoring the microscopic details of stepping on filaments, coarse-grained random walk models have been employed * shaebani@lusi.uni-sb.de to study the two-state dynamics of molecular motors [19][20][21][22]. Dentritic immune cells also move persistently (migration phase) interrupted by slow phases for antigen uptake [23]. There have been many other locomotive patterns in biological and non-living systems investigated via models with distinct states of motility [24][25][26][27][28][29][30][31]. For instance, the problem of searcher proteins finding a specific target site over a DNA strand has been studied by multi-state stochastic processes [32][33][34].
The particle trajectories obtained from experiments often comprise a set of recorded positions of the particle, from which the successive directions of motion can be deduced. These directions are correlated on short time scales for active motions. However, the trajectory eventually gets randomized and the asymptotic dynamics is diffusive, with a diffusion constant D asymp that depends on the particle velocity and persistency [35,36]. One expects a similar long-term behavior for a mixture of run and tumble dynamics as well. The question arises how the transient short time dynamics, the crossover time to asymptotic diffusion, and D asymp depend on the run and tumble velocities and the switching probabilities between the two states. It is also unclear how the overall dynamics is influenced by the choice of the initial conditions of motion, like the probabilities to start either in the run or tumble state, which are parameters that can be extracted from experimental data.
Here, we present a two-state model for the run-andtumble dynamics with spontaneous switchings between the states of motility. By deriving an analytical expression for the time evolution of the mean square displacement, we show how the interplay between the run and tumble velocities, the transition probabilities, and the initial conditions of motion leads to various anomalous transport regimes on short and intermediate time scales. We particularly clarify how the probability of starting from run or tumble state diversifies the transient anomalous regimes of motion, and verify that the long-term diffusion constant D asymp does not depend on the choice of the initial conditions of motion.

II. MODEL
We develop a stochastic model for the run-andtumble dynamics with spontaneous transitions between the motility states. We consider a two-state random walk in discrete time and continuous space with the following characteristics: The run phase is a persistent random walk with persistency p and mean speed v R . The dynamics in the tumble phase is an ordinary diffusion with the mean speed v T . The asymmetric transition probabilities from run to tumble phase and vice versa are denoted, respectively, by f R→T and f T→R . As a result of constant transition probabilities, the run and tumble times are exponentially distributed in our model. This restriction can be relaxed by introducing time-dependent transition probabilities [37]. To characterize the persistency of the run phase, we use the probability distribution F R (θ) of directional changes along the trajectory in the run phase. The directional persistence can be characterized by the persistency parameter p = π −π dθ e iθ F R (θ), which leads to p= cos θ for symmetric distributions with respect to the arrival direction. Thus, p ranges from 0 for pure diffusion to 1 for ballistic motion and reflects the average curvature of the run trajectories. Similarly, we define F T (θ) for the probability distribution of directional changes along the trajectory in the tumble phase, and F R→T (θ) and F T→R (θ) for the directional changes when switching between the states occurs [see Fig. 1(A)]. In the tumble phase (i.e. an ordinary diffusion), the probability distribution of directional changes is isotropically distributed (F T (θ)= 1 2π ), leading to a zero persistency. The run-and-tumble stochastic process can be described in discrete time by introducing the probability densities P R t (x, y|α) and P T t (x, y|α) to find the particle at position (x, y) arriving along the direction α at time t in the run and tumble states, respectively. α is defined with respect to a given reference direction, as shown in Fig. 1(B). Denoting the time interval between consecutive recorded positions of the particle by ∆t, the following set of master equations describe the dynamical evolution of the probability densities (1) Each of the two terms on the right-hand side of the equations represents the possibility of being in one of the two states in the previous time step (see Fig. 1(B) for the particle trajectory during two successive steps). The probability of starting the motion in the run or tumble phase is denoted by P R 0 and P T 0 , respectively (with P T 0 =1−P R 0 ). The change in the direction of motion θ=α−γ with respect to the arrival direction is randomly chosen according to the turning-angle distribution F R (θ) or F T (θ) in the run or tumble state, respectively. Both distributions are symmetric with respect to the arrival direction (i.e. left-right symmetric in 2D). We assume for simplicity that the directional change during the transition between the two states follows the turning-angle distribution of the new state, corresponding to F R→T (θ)=F T (θ) and F T→R (θ)=F R (θ). However, in general one should consider independent turning-angle distributions with nonzero mean for F R→T (θ) and F T→R (θ) as, for instance, a sharp change in the direction of motion of E. coli or Bacillus Subtilis when switching from tumbling to running is observed [1,2,6].
The total probability density P t+∆t (x, y|α) to find the particle at position (x, y) arriving along the direction α at time t+∆t is given by P t+∆t (x, y|α) = P R t+∆t (x, y|α)+ P T t+∆t (x, y|α). Using the Fourier transform of the probability density in each state h (h∈{R, T }), defined as the Fourier transform of the total probability density is given by P t+∆t (k|m) = P R t+∆t (k|m) + P T t+∆t (k|m), from which the moments of displacement can be calculated as By means of a Fourier-z-transform technique, it is possible to solve the master equations (1) to obtain the time evolution of the moments of displacement [36,38,39].
Here we briefly explain the procedure to calculate the mean square displacement (MSD) as the main quantity of interest. From Eq. (3), the MSD is given as where (k, φ) is the polar representation of k. As- Next we apply the Fourier transformation on the master equations (1). For example, the first master equation after Fourier transform reads Then by using the qth order Bessel's function replacing e ikv R ∆t cos(α−φ) with π −π dβe ikv R ∆t cos β δ(β−(α−φ)), and using it follows that P R t+∆t (k, φ|m) can be expanded as a Taylor series We expand both sides of Eq. (6) and collect all terms with the same power in k. As a result, recursion relations for the Taylor expansion coefficients can be obtained. For instance, for the terms with power 0 in k one finds Similarly, the expansion coefficients of terms with higher powers in k can be calculated and the procedure is repeated for the second master equation in (1). As a result, a set of coupled equations is obtained for each expansion coefficient, connecting time steps t+∆t and t. Applying a z-transform Q(z)= ∞ t=0 Q t z −t enables one to solve these sets of equations. Particularly the coefficients of terms with power 2 in k, i.e. Q R 2 (z, φ|m) and Q T 2 (z, φ|m), are useful to calculate the MSD Finally we obtain the following exact expression for the MSD in z space where By inverse z-transforming Eq. (10), the MSD can be obtained as a function of time. The resulting general expression for the MSD r 2 (t)=2 x 2 (t) is lengthy and depends on the run persistency p, the speeds v R and v T , the transition probabilities f R→T and f T→R , and the probability of initially starting in the run P R 0 or tumble state P T 0 =1−P R 0 . r 2 (t) typically consists of linear and exponentially decaying terms with t as well as time-independent terms, as shown in the following in the special case of constant velocity and the initial condition of starting in the run state. By choosing ∆t=1, v R =v T =1, and the initial condition P R 0 =1, the general expression of r 2 (t) reduces to

III. RESULTS AND DISCUSSION
We first investigate the time evolution of the MSD for different values of the key parameters p, v R , v T , f R→T , f T→R , and P R 0 . As a simple check, the expression (10) for f R→T =0, f T→R =1, v T =0 and P R 0 =1 reduces to and by inverse z-transforming, the MSD for a single-state persistent random walk [35,40] x 2 (t) = is recovered. In Fig. 2   time scales leading to the observed slow dynamics in this regime. The asymptotic dynamics is however diffusive since the linear term eventually dominates. It can also be seen that the crossover time to asymptotic diffusion varies by several orders of magnitude upon changing the parameter values. The crossover time can be characterized as the time at which the exponentially decreasing terms in x 2 (t) become smaller than the terms which survive at long times. We find that the convergence of the MSD to its asymptotic diffusive form can be described by the sum of two exponential functions with the characteristic times . The prefactors B 1 and B 2 are func- Figure 3 shows how the characteristic times t c1 and t c2 vary upon changing the key parameters. Although the slopes of the exponential decays in Eq. (14) are solely determined by the transition probabilities f R→T and f T→R and the run persistency p, the crossover time to the asymptotic diffusive dynamics is also influenced by the other dynamic parameters of the model through the prefactors B 1 and B 2 . For example, for the set of parameter values p=0.9, v R =10, v T =0.1, f R→T =0.1 and f T→R =0.01, the convergence time (with 5% accuracy) to the asymptotic dynamics for P R 0 =1 is nearly twice as long as for P R 0 =0. Figure 2 also shows that the asymptotic diffusion constant D asymp varies by changing the key parameters. The differences in the y-intercept of the dashed (asymptotic) lines in log-log plots reflect the sensitivity of D asymp to the model parameters. By inverse z-transforming of Eq. (10) and taking the limit t→∞, we obtain D asymp (i.e. the coefficient of the term linear in time) in the general form as While the diffusion coefficient trivially increases with the speed, its dependency on f R→T , f T→R and p is more complicated and shown in Fig. 4. D asymp varies by several orders of magnitude as a function of these parameters. Under the specific condition F R→T (θ)=F T (θ)=δ(θ) and F T→R (θ)=F R (θ) and v T =0, the walker stops when entering the tumble phase without changing its arrival direction and it returns smoothly to the run phase without experiencing a kick (i.e. a sharp change in the direction of motion). Motor-driven transport along cytoskeletal filaments in crowded cytoplasm exhibits such a run-andpause dynamics [21,41]. In this case, one obtains In the limit p→1 the trajectory becomes nearly straight implying that the randomization time and the covered area until reaching the asymptotic diffusive regime (and thus D asymp ) diverge. Interestingly, D asymp in Eq. (15) is independent of P R 0 and P T 0 , i.e. the initial condition of starting the motion in the run or tumble state. Thus the analytical results predict that the asymptotic diffusive dynamics, characterized by the linear time-dependence does not depend on the initial conditions. In Fig. 5 we present the simulation results for several values of P R 0 . At long times, all curves merge and follow the analytical prediction Eq. (17). Note that only the linear term in time is independent of P R 0 and the exponentially decaying and time-independent terms in the MSD depend on the initial conditions [see e.g. Eq. (11)]. Although the process is Markovian it keeps initially for some time its memory of the initial direction and state of motion. However, the influence of the P R 0 -dependent terms vanishes in the limit t→∞ and the time dependence of the MSD appoaches the asymptotic linear form Eq. (17).
The short time dynamics is, however, strongly influenced by the choice of the initial conditions of motion. Figure 5 shows that the initial slope of the MSD curve varies with P R 0 . One can assign an initial anomalous exponent κ to the MSD curve by fitting the power-law r 2 ∼t κ . By choosing the first two data points of the MSD curve, the fitting leads to r 2 (t=2)/ r 2 (t=1)=2 κ . Thus, the initial anomalous exponent κ can be deduced from the MSD at t=1, 2 as κ = ln r 2 (t=2) After replacing the MSD at t=1, 2 obtained from Eq. (10) we get Figure 6(A) shows how the initial conditions of motion influences the initial anomalous exponent for a given set of parameters. Note that the displayed monotonic growth of κ with P R 0 does not hold in general, as we observed decreasing as well as nonmonotonic functionalities by vary-ing other parameter values. However, κ increases monotonically with p in all parameter regimes as shown in Fig. 6(B). Moreover, figures 6(C) and (D) show that κ also varies widely with the speed and transition probabilities. Because of combining an active run state (0<p<1) and normal diffusion (tumble state), κ remains above 1 (superdiffusion). However, by generalizing the run state to include subdiffusive motion (i.e. when −1<p<1), κ can decrease below 1.
To better understand the role of the initial conditions, we note that the steady probabilities P R steady and P T steady of finding the particle in each of the two states is determined by the transition probabilities f R→T and f T→R . Therefore, the influence of the initial condition of starting the motion in any of the two states gradually vanishes as the probabilities P R (t) and P T (t) of finding the particle in the run or tumble state gradually approach their steady values. By considering a discrete time Markov process with transition probabilities f R→T and f T→R , the probabilities at time t can be obtained from those at time t−1 as (20) By applying this relation recursively, one can derive the probabilities at time t based on the initial probabilities Thus the evolution of P R (t) and P T (t) obeys . If one starts with the initial condition P R 0 =P R steady , the system is immediately equilibrated. Otherwise, the choice of the initial conditions of motion affects the short-time dynamics and diversifies the transient anomalous diffusive regimes. According to Eq. (22), the relaxation of the probabilities toward their steady values follow an exponential decay P R (t), P T (t) ∼ e −t/tm with t m = −1 ln |1−f R→T −f T→R | . While the characteristic time for the relaxation of the probabilities solely depends on the transition probabilities, the characteristic time for the crossover to asymptotic dynamics is influenced additionally by the run persistency, as we showed previously in Eq. (14). Therefore, there are two independent relaxation times t m (=t c1 ) and t c2 . In case the relaxations occur on different time scales, two distinct crossovers in the time evolution of the MSD may be observed in general as can be seen in Fig. 2(A).

IV. CONCLUSION
We presented a persistent random walk model to study the stochastic dynamics of particles with active fast and passive slow motility modes. We derived an exact analytical expression for the mean square displacement, which allows one to analyze the transient anomalous transport regimes on short time scales and also extract the characteristics of the asymptotic diffusive motion such as the crossover time and the long-term diffusion constant. In particular we showed that while the choice of the initial conditions of motion influences the anomalous diffusion at short times, the asymptotic behavior remains independent of it and is entirely controlled by the run persistency, the velocities of the run and tumble states and the transition probabilities between the two states.