Optimization and universality of Brownian search in quenched heterogeneous media
The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time, the so called mean first passage time (MFPT) problem. The occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the MFPT, by reducing oversampling of space. Here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension. In a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piece-wise constant diffusivity. The MFPT is analyzed under the constraint of conserved average dynamics, that is, the spatially averaged diffusivity is kept constant. Our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the MFPT to the target. We prove that the MFPT for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality. In contrast to intermittent strategies, which are most profitable in low spatial dimensions, the spatially inhomogeneous search performs best in higher dimensions. Discussing our results alongside recent experiments on single particle tracking in living cells we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes.
Random search processes are ubiquitous in nature (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15); (16); (17), ranging from the diffusive motion of regulatory molecules searching for their targets in living biological cells (5); (8); (18); (19); (20); (21); (22); (23); (24); (25), bacteria and animals searching for food by active motion (2); (8), all the way to the spreading of epidemics and pandemics (3); (4) and computer algorithms in high-dimensional optimization problems (26). The fact that the search strategy in these processes is to a large extent random reflects the incapability of the searcher to keep track of his past explorations at least over more than a certain period (5). During the years several different search strategies have been studied in the literature, including Brownian motion (27); (18); (8); (28), spatio-temporally decoupled Lévy flights (LFs) (29); (30); (31); (32); (33); (34) and coupled Lévy walks (LWs) (35); (36); (37); (38); (39); (40); (41); (42); (43); (44); (45); (46) in which the searcher undergoes large re-allocations with a heavy-tailed length distribution either instantaneously (LFs) or with constant speed (LWs), as well as intermittent search patterns, in which the searcher combines different types of motion (5), for instance, three-dimensional and one-dimensional diffusion (18); (19); (20); (21); (22); (23); (24); (25); (47); (48), three-dimensional and two-dimensional diffusion (49); (50), or diffusive and ballistic motion (5); (8); (51); (52); (53); (54); (55); (56); (57).
The efficiency of the search strategy is conventionally quantified via the mean first passage time (MFPT) defined as the average time a random searcher needs to arrive at the target for the first time (27); (28); (5); (58); (59). The physical principle underlying an improved search efficiency is an optimized balance between the sampling of space on a scale much larger than the target and on the scale similar to or smaller than the target (5). More specifically, periods of less-compact exploration—for instance, diffusion in higher dimensions, Lévy flights, or ballistic motion—aid towards bringing the searcher faster into the vicinity of the target. Concurrently a searcher in such a less compact search mode may thereby easily overshoot the target (32); (33). In contrast, compact exploration of space (for instance, diffusion in one dimension) is superior when it comes to hitting the target from close proximity but performs worse when it comes to the motion on larger scales taking the searcher from its starting position into the target’s vicinity: typically frequent returns occur to the same location, a phenomenon referred to as oversampling. Mathematically this is connected to the recurrent nature of such compact random processes. The idea behind search optimization is to find an optimal balance of both more and less compact search modes in the given physical setting (5). For instance in the so-called facilitated diffusion model for the target search of regulatory proteins on DNA(19); (20); (21); (22); (23) the average duration of non-compact three-dimensional free diffusion is balanced with an optimal compact one-dimensional sliding regime along the DNA molecule. In intermittent search (5); (8); (51) persistent ballistic excursions are balanced by compact Brownian phases. This optimization principle intuitively works better in lower dimensions, where a searcher performing a standard random walk oversamples the space. Hence, the typically considered optimized strategies have the largest gain in low dimensions.
In a variety of experimental situations the motion of a searcher is characterized by the same search strategy but is not translationally invariant. A typical example is a system in which the searcher performs a standard random walk but with a spatially varying rate of making its steps. This type of motion is actually abundant in biological cells, where experiments revealed a distinct spatial heterogeneity of the protein diffusivity (60); (61); (62). Several aspects of such diffusion in heterogeneous media have already been addressed (27); (63); (64); (65), but the generic FPT properties remain elusive, in particular, for quenched environments.
Here was ask the question whether spatial heterogeneity is generically detrimental for the efficiency of a random search process or whether it could even be beneficial. Could it even be true that proteins find their targets on the genome in the nucleus faster because their diffusivity landscape in the cell is heterogeneous? On the basis of exact results for the MFPT in one, two, and three dimensions in a closed domain under various settings we here show that a spatially heterogeneous search can indeed significantly enhance the rate of arrival at the target. We explain the physical basis of this acceleration compared to a homogeneous search process and quantify an optimal heterogeneity, which minimizes the MFPT to the target. Furthermore we show that heterogeneity can be generically beneficial in a random system and is thus a robust means of enhancing the search kinetics. The optimal heterogeneous search rests on the remarkable observation that the MFPT is completely dominated by those trajectories heading directly towards the target. We prove that the MFPT for the heterogeneous system can be exactly described with the results of a standard random walk. We compare our theoretical findings to recent experiments on single particle tracking in living cells, which are indeed in line with the requirements for enhanced search.
The paper is organized as follows. Section II introduces our minimal model for heterogeneous search processes. In Sec. III we briefly summarize our main general results, which hold irrespective of the dimension (, , or ). Section IV is devoted to the analysis of the most general situation with a specific starting point and position of the interface. In Sec. V we focus on the Global MFPT, that is, the MFPT averaged over the initial position. In Section VI we analyze a system with a random position of the interface and optimize the MFPT averaged over the interface position. In Section VII we address the Global MFPT in systems with a random position of the interface. Throughout we discuss our results in a biophysical context motivated by recent experimental findings. Finally, we conclude by discussing the implications of our results for more general spatially heterogeneous systems.
Ii Minimal model for spatially heterogeneous random search
We focus on the simplest scenario of a spatially heterogeneous system. Even for this minimal model the analysis turns out to be challenging and our exact results reveal a rich behavior with several a priori surprising features. We consider a spherically symmetric system in dimensions , , and with a perfectly absorbing target of radius located in the center (Fig. 1). The outer boundary at radius is taken to be perfectly reflecting. The system consists of two domains with uniform diffusivities denoted by and in the interior and exterior domains, respectively. The interface between these domains is located at . The microscopic picture we are considering corresponds to the ’kinetic’ interpretation of the Langevin or corresponding Fokker-Planck equations. In particular we assume that the dynamics obey the fluctuation-dissipation relation and in the steady state agree with the results of equilibrium statistical mechanics (66).
In the biological context we consider that the system is in contact with a heath bath at constant and uniform temperature . The signaling proteins diffuse in a medium comprising water and numerous other particles, such as other biomacromolecules or cellular organelles. The remaining particles, which we briefly call crowders, are not uniformly distributed across the cell—their identity and relative concentrations differs within the nucleus and the cytoplasm and can also show variations across the cytoplasm. The proteins hence experience a spatially varying friction , which originates from spatial variations in the long-range hydrodynamic coupling to the motion of the crowders, which is in turn mediated by the solvent (67); (68). The proteins thus move under the influence of a position dependent diffusion coefficient and a fluctuation-induced thermal drift (see (69) for details). The vital role of such hydrodynamic interactions in the cell cytoplasm was demonstrated in (70).
Because of the spherical symmetry of the problem we can reduce the analysis to the radial coordinate alone. That is, we only trace the projection of the motion of the searcher onto the radial coordinate) and therefore start with a discrete space-time nearest neighbor random walk in-between thin concentric spherical shells of equal width as depicted in Fig. 1. The shell denotes the region between surfaces with radii and . We assume that the hopping rates between shells obey detailed balance
|Here denotes the probability distribution of finding the particle in shell . The hopping rates are given as the product of the intrinsic rate and , the probability to jump from shell to shell (and for jumps in the other direction),|
|Here is the arithmetic mean of the diffusivity in shell . The rate can be derived as follows. A random walker located in shell at time can either move to shell with probability or to shell with probability . Due to the isotropic motion of the random walker these probabilities are proportional to the respective surface areas of the bounding -dimensional spherical surfaces at and . That is, and . The proportionality constant is readily obtained from the normalization condition leading to|
and thus . Therefore, while the random walker moves in all directions (radial, azimuthal, or polar) we can project its motion on the radial coordinate only. We assume that the interface is located between two concentric shells leading to a continuous steady state probability density profile. The searcher starts at uniformly distributed over the surface of a -sphere with radius , as sketched in Fig. 1a).
In our analytical calculations we model the system in terms of the probability density function to find the particle at radius at time after starting from radius at . obeys the radial diffusion equation
|with piece-wise constant diffusivity|
|We assume that the target surface at radius is perfectly absorbing,|
|to determine the first passage behavior. At the outer radius we use the reflecting boundary condition|
These boundary conditions are complemented with joining conditions at by requiring the continuity of the probability density and the flux, which follow from our microscopic picture.
To quantify our model system we introduce the ratio
|of the inner and outer diffusivities. Moreover, we demand that the spatially averaged diffusivity|
remains constant for varying and . Without such a constraint the problem of finding an optimal , which minimizes the MFPT is ill-posed and has a trivial solution . More importantly, we want to compare the search efficiency as a function of the degree of heterogeneity, where the overall intensity of the dynamics is conserved. Returning to our microscopic picture of a signaling protein searching for its target in the nucleus, the heterogeneous diffusivity is due to spatial variations in the long-range hydrodynamic coupling to the motion of the crowders. Their identity and relative concentration in the cell varies in space, but the effect is mediated by thermal fluctuations in the solvent at a constant temperature. The constraint in Eq. (3b) then corresponds to a redistribution of the crowders at constant temperature, cell volume and numbers of the various crowders, which would not affect the spatially averaged diffusivity.
Under the constraint (3b) of constant spatially averaged diffusivity we obtain for any given and that
|where we introduced the hypervolume ratio|
of the inner versus the entire domain excluding the target volume. To solve Eq. (2a) we take a Laplace transform in time and the obtained Bessel-type equation is solved exactly as shown in Sec. IV. The MFPT for the particle to reach the target surface at is obtained from the Laplace transformed flux into the target
|via the relation|
The angular prefactor for , for and for . We treat the degree of heterogeneity as an adjustable parameter at a fixed value of and seek for an optimal value minimizing the MFPT. The optimal heterogeneity, which we denote with an asterisk, is thus obtained by extremizing with respect to .
Iii Summary of the main results
Since in combination with the diffusivity sets the absolute time scale we can express, without loss of generality, time in units of , set and focus on the problem in a unit sphere. We introduce dimensionless spatial units , and . For the sake of completeness, we retain the explicit and dependence in the prefactors.
Our first main result represents the fact that the MFPT to the target in the inhomogeneous system in dimension , and , can be expressed exactly in terms of the corresponding MFPT in a homogeneous system with diffusivity , with or . Remarkably the MFPT is thus exactly equal to
In the second line the argument stands for the release of the particle at the interface, and the index of the last term is used to indicate that this term measures the first passage to the interface at . That is, in this case when the particle starts in the inner region with diffusivity Eq. (6) reveals that the MFPT of the heterogeneous system is equal to that of a homogeneous system with diffusivity everywhere and is independent of the position of the interface. Conversely, if the searcher starts in the exterior region with diffusivity the MFPT contains two contributions: (i) the MFPT from to in a homogeneous system with diffusivity and (ii) the MFPT from to in a homogeneous system with diffusivity , as shown schematically in Fig. 2. Eq. (6) is exact and independent of the choice for and thus holds for an arbitrary set of diffusivities and even if . In other words, it is not a consequence of a conserved .
The additivity principle of the individual MFPTs in Eq. (6) is only possible if the excursions of the searcher in the directions away from the target are statistically insignificant. We would expect that some trajectories starting in the inner region will carry the searcher into the outer region with diffusivity before they eventually cross the interface and reach the target by moving through the inner region with diffusivity . Such trajectories will obviously be different in the heterogeneous system in comparison to a homogeneous system with diffusivity everywhere, This appears to contradict the complete independence of of the MFPT in Eq. (6) for trajectories with . This observation can be explained by the dominance of direct trajectories, whose occupation fraction outside the starting radius is statistically insignificant: The MFPT for a standard random walk in dimensions , , and is completely dominated by direct trajectories. As such, the MFPT is really a measure for the efficiency of the fast trajectories.
Our second main result demonstrates that for a finite optimal heterogeneity exists at given interface position and is given by
For this value the MFPT attains a minimum. Hence, the optimal heterogeneity is completely determined by the volume fractions and the MFPT properties and hence strictly by the direct trajectories. As above, the index in the MFPT indicates the first passage to the interface, while without this index the MFPT quantifies the first passage to the target at . The explicit results for are shown in Fig. 4 and are discussed in detail in Sec. IV.
Often one is interested in the MFPT averaged over an ensemble of starting positions, the Global MFPT . As before, it can be shown that an optimal heterogeneity exists for any interface position and is universally given by
Similar to the general case is again proportional to but here the corresponding MFPTs in the second factor are reduced by a spatially averaged change of the MFPT with respect to the starting position. The optimal heterogeneity for the Global MFPT is shown in Fig. 5, and discussed in detail in Sec. V.
In a setting when the interface position is random and uniformly distributed we are interested in the MFPT from a given starting position averaged over the interface position. A measurable quantity for this scenario for an ensemble of random-interface systems is the MFPT , where the curly brackets denote an average over the interface positions . Explicit results for dimensions , , and are given in Sec. VI. Solving for the optimal heterogeneity we obtain
The optimal heterogeneity for the MFPT averaged over the random interface position is shown in Fig. 6.
Finally, we compute the Global MFPT averaged over the position of the interface, , whose explicit results are given in Sec. VI. Also here an optimal strategy can be identified as
The optimal heterogeneity for the Global MFPT averaged over the random interface position is shown in Fig. 7.
Eq. (6) represents a rigorous proof that direct trajectories dominate the MFPT for Brownian motion. In addition, the heterogeneity does not affect the fraction of direct versus indirect trajectories but only their respective durations. Due to the fact, that indirect trajectories are statistically insignificant, we can in principle make them arbitrarily slow if we start in the inner region. But as we let the inner diffusivity go to infinity (and hence the outer one to zero) we are simultaneously slowing down the arrivals to the interface if starting from the outer region. The physical principle underlying the acceleration of search kinetics is: The optimal heterogeneity corresponds to an improved balance between the MFPT to reach the interface and the one to reach the target from the interface.
Iv Mean first passage time for fixed initial and interface positions
Here we present the mathematical derivation and the explicit results for the situation with a specific initial condition and interface position . Eq. (2a) is solved by Laplace transformation, and the resulting Green’s function with the boundary conditions (2c) and (2d) reads
|where we introduced the abbreviation and the auxiliary functions|
Here the and denote the modified Bessel functions of order of the first and second kind, respectively. The Laplace transformed first passage time density is obtained from the flux (5a) into the target, and the MFPT then follows from relation (5b). In case of the target size only enters the problem by determining the width of the interval. Using it can be shown that Eq. (11a) reduces to the ordinary expression for regular diffusion given in Ref. (27) for and .
Here the left index denotes the dimensionality of the system. Our results show excellent agreement with numerical simulations, as demonstrated in the Appendix. Plugging the above expressions (12) into Eq. (6) we can compare the search efficiency with respect to a homogeneous random walk by introducing the dimensionless ratio
The corresponding results for dimensions , 2, and 3 are shown in Fig. 3.
The qualitative behavior of the MFPT with respect to —that is, the degree of the heterogeneity—depends on the starting position relative to the interface. If the initial position lies in the inner region decays monotonically and saturates at a finite asymptotic value, . Hence, in this case an optimal strategy does not exists and the best search performance is achieved for large diffusivities in the inner region. The lower bound on is set by the volume fraction of the inner region, which sets a bound on the ratio . This result is yet another manifestation of the fact that the MFPT is completely dominated by direct trajectories.
Conversely, if the searcher starts in the outer region an optimal strategy exists. To understand the existence and meaning of the optimal heterogeneity we perform a power series expansion of . We find the scaling as , which is due to the fact that in this regime , which dominates the MFPT in this regime. In the other limit as we find because here and the rate determining step is the arrival at the interface. We can understand the optimal heterogeneity as a beneficial balance between the rate of arriving at the interface from the starting position and the rate to find the target if starting from the interface.
In contrast, too large values of prolong the time to reach the interface and cannot be compensated by a faster arrival from the interface towards the target. The optimal heterogeneity as a function of the starting position is shown in Fig. 4a)-c) and reveals the divergence as : this point corresponds to the disappearance of an optimal strategy. As gradually approaches unity, continuously decreases towards a plateau, which suggests that to reach the target from the interface becomes the rate determining step. Moreover, decreases with increasing target size, because the inner region becomes smaller and thus allows a larger in the optimal scenario. The overall gain of an optimal search with respect to a standard random walk is shown in Fig. 4d)-f). Only in the domain an optimal heterogeneity exists, therefore we omitted the region . As mentioned before, we observe the monotone convergence from above as and hence the heterogeneous search always outperforms the standard Brownian search for . The highest gain is therefore set by the volume fraction of the inner region which becomes arbitrarily small as . For we find that the gain is largest for starting positions near the interface and when the interface is not too close to the outer boundary, where the system essentially approaches the homogeneous limit. The variations are larger in higher dimensions, which is, of course, connected with the pronounced spatial oversampling in lower .
Different from conventional strategies—intermittent or Lévy-stable processes, which affect the compactness of exploring space—heterogeneous search is more profitable in higher dimensions (see Fig. 4d)-f)). Because heterogeneity acts by enhancing/retarding the local dynamics and does not affect spatial oversampling it is intuitive that it performs better for non-compact exploration of space. Both the existence as well as the gain of an optimally heterogeneous search are thus a direct consequence of direct trajectories dominating the MFPT.
V Global mean first passage time for fixed interface position
The Global MFPT is obtained by direct averaging of Eq. (6) over the initial position ,
The exact expressions for the Global MFPT in the various dimensions read
These are to be compared with their homogeneous counterparts
in , , and , respectively. As before we introduce the dimensionless enhancement ratio
The result are shown in Fig. 5. In this case we are effectively considering a weighted average of the results presented in the previous Sec. IV. Noticing that most of the volume of a -sphere is increasingly concentrated near the surface for growing dimensions we anticipate that the results will be more prominently influenced by the features of trajectories starting further away from the target. An interesting question will therefore be whether an optimal heterogeneity exists for all interface positions and dimensions. An expansion of in a power series of reveals that as and as for all positions of the interface (see Fig. 5 a)-c)). Thus, an optimal heterogeneity always exists for all . From Fig. 5 a)-c) we observed that the dependence of the relative gain compared to the on the interface position in the limit is very weak in all dimensions as long as it is not too close to the external boundary. In the case of large somewhat larger variations are observed for dimensions and 2. In the dependence on the interface position is largest near the optimal value for but very weak elsewhere. Hence, a control of the target location dynamics by adjusting is only efficient near the optimal point . This is observed from the respective scaling and as and as explained in Sec. IV. Overall the gain with respect to the homogeneous random walk is larger for higher dimensions, which has the same origin as in the general case discussed above, however here the additional effect of averaging over the initial position enters. The optimal heterogeneity has the exact results
The behavior of Eqs. (18) is shown in Fig. 5e)-f). In higher dimensions the optimal heterogeneity shows a non-monotonic behavior with respect to the interface position and increases upon approaching the target or the outer boundary. Simultaneously, the overall dependence of the Global MFPT on vanishes in these limits—see the inset of Fig. 5)e)-f)—corresponding to the situation when the system is no longer heterogeneous. These results can be rationalized by the fact that in the limit the ratio becomes negligible and hence a higher is allowed without slowing down the dynamics of reaching the interface from the external region. Conversely, as also and smaller are allowed because the rate limiting step is hitting the target from the inner region. In both limits, however, the overall enhancement effect with respect to a standard random walk becomes negligible. Away from these limits the gain of the optimal heterogeneity is larger for higher dimensions and can be remarkably large for intermediate interface positions (see the inset of Fig. 5)e)-f) and increases with decreasing target size.
In a biological context, the present setting is relevant for signaling proteins searching for their target in the nucleus when the proteins are initially uniformly distributed throughout the cell cytoplasm. Recent experiments revealed a significant heterogeneity in the spatial dependence of the protein diffusion coefficient across the cell, with a faster diffusivity near the nucleus (60); (61). Such a spatial heterogeneity could therefore be beneficial for the cell by accelerating the dynamics of signaling molecules.
Vi Mean first passage time in a random heterogeneous system for fixed initial position
We now address the MFPT problem when the interface position is random in a given realization and uniformly distributed over the radial domain. Specifically, we are interested in the MFPT of a particle starting at averaged over the interface position ,
Experimentally, this would correspond to measuring an ensemble of systems with random value of . For dimensions , , and the MFPT has the explicit form
The gain compared to the homogeneous random walk,
is shown in Fig. 6a)-c) and reveals a somewhat less sharp optimal heterogeneity as compared to the Global MFPT. The limiting behavior for small and large is the same as for the previous cases but here there is a wider range of values producing a comparable gain. We also observe a stronger dependence on the starting position as and a very weak dependence for , which is yet another consequence of direct trajectories dominating the MFPT. The optimal heterogeneity is obtained from Eq. (9) and reads
Here we introduced the auxiliary functions
The results for various dimensions are depicted in Fig. 6d)-6f). Accordingly an optimal heterogeneity always exists. Note that the dependence of on the target size in and 3 becomes reversed upon increasing the initial separation to the target. When starting near the target the initial position will on average lie in the inner region in an ensemble of realizations of the interface position. The optimal heterogeneity will thus on average correspond to a very fast diffusion in the inner region. For small the probability of starting in the inner region, , will be lower for smaller targets and hence will be larger accordingly. Conversely, if starting further away from the target will on average lie in the outer region and the search time will be more strongly influenced by the rate of arriving at the interface in each realization. An optimal heterogeneity will therefore correspond to a smaller asymmetry of diffusivities in the inner and outer regions. More specifically, since the probability of starting in the outer region, , will be lower for larger targets the value of will accordingly be smaller. The gain of the optimal heterogeneity is shown in the insets of Fig. 6d)-6f) and reveals a significant improvement with respect to the standard random walk in every dimension. The dependence on target size has again a non-monotonic behavior, which follows from the argument presented above. Hence, even in a system with a random and quenched position of the interface a spatially heterogeneous search process will on average strongly outperform the standard random walk in every dimension.
In a biological context the present setting is important for the experimentally observed quenched spatial disorder revealed by single-particle tracking (62); (65); (71); (61); (72); (73). More generally, the ideas can also be extended to the dynamics in complex disordered systems (74); (75) and Sinai-type diffusion (76). Our results show that quenched spatial heterogeneity can robustly enhance random search processes even in a disordered setting. This robustness could eventually be exploited in search strategies, because it requires less prior knowledge about the location of the target.
Vii Global mean first passage time in a random system
This section completes our study, addressing the Global MFPT in a random system,
As far as blind spatially heterogeneous search is concerned this is the most robust setting. It can be shown that the exact results for have the form