In Lévy walks (LWs), particles move with a fixed speed along straight line segments and turn in new directions after random time intervals that are distributed according to a power law. Such LWs are thought to be an advantageous foraging and search strategy for organisms. While complex nervous systems are certainly capable of producing such behavior, it is not clear at present how single-cell organisms can generate the long-term correlated control signals required for a LW. Here, we construct a biochemical reaction system that generates long-time correlated concentration fluctuations of a signaling substance, with a tunable fractional exponent of the autocorrelation function. The network is based on well-known modules, and its basic function is highly robust with respect to the parameter settings.
The migration paths of microorganisms can often be modeled as random walks. The simplest case of a (short-term) correlated random walk is the Ornstein-Uhlenbeck process, in which the autocorrelation function1 of the momentaneous velocity decays exponentially . By contrast, in Lévy walks (LWs), particles move with a fixed speed, and turn abruptly after random time intervals that are distributed according to a power law (PL) , thereby leading to long-term correlated trajectories. Such LWs are thought to be an advantageous foraging and search strategy for the organisms [1,,,–5, 7, 8, 11, 14, 16, 17, 20, 21]. However, it is not clear at present how single-cell organisms without complex nervous systems can generate the long-term correlated control signals required for a LW. Here, we show that such signals can arise simply by the combination of a Poisson point process with a positive feedback loop and a linear degradation mechanism, ingredients found frequently in even the simplest biochemical pathways .
We construct a chemical reaction system that generates random concentration peaks of a signaling substance, which could be used to trigger the different motor actions of a Lévy walker. We demonstrate that for a wide range of system parameters the intervals between trigger events are distributed according to a power law with fractional and tunable exponent γ, the hallmark of long-term correlations.
In what follows, we assume a well-stirred reaction system of unit volume, so that (continuous) concentrations and (integer) molecule numbers can be used interchangeably. We aim to design a reaction network that spontaneously creates a sequence of concentration pulses of a signaling substance B with a power law distribution p(τ) ∝ τ−γ of pulse durations. The autocorrelation of such a pulse train is long-time correlated with a fractional exponent in the range [−1, 0]. Concentration pulses of a given duration τ can be realized by quickly raising the concentration NB(t) of the signaling substance from NB(t1) ≈ 0 to a value NB(t2) ∝ τ that is proportional to the desired pulse duration, and thereafter decreasing NB(t) at a constant rate Rdeg, until it reaches zero again at time t3 (compare Figure 1).
Using this scheme, we start from a small concentration NB(t1) of the signaling substance B at time t1 and let it grow exponentially. At time t2, the growth period of NB(t) is stopped by a Poisson point process. This way, the growth time t2 − t1 is exponentially distributed, and the combination with the exponential growth of NB(t) results in a power law distribution of the peak concentration NB(t2).
Chemically, the exponential growth of NB(t) is achieved via an autocatalytic reaction that has the property . The Poisson point process is naturally realized by the diffusive arrival or disappearance of a diluted molecular species at the reaction site. The detailed chemical equations of our proposed reaction network are presented in the following section.
3 Detailed Reaction Network
3.1 Generation Reactions
We have thus shown that NB is increasing exponentially during the growth period. If the duration t of this period could be randomly drawn from an exponential distribution, then at the end of the growth period the molecule number NB would have the desired power law distribution.
3.2 Decay Reactions
3.3 Switch Reactions
The four chemical reactions discussed above are not yet sufficient for proper function of repeated pulse generation.
On the one hand, the molecule species D disturbs the autocatalytic reaction (2): If some molecules B already decay by reacting with D during the generation phase, then NB does not grow exponentially.
3.4 Input Reactions
3.5 Output Reactions
Since we have shown that the durations of the linear decay (running) phases are distributed according to a tunable power law, (compare Equation 1), this results in a Lévy walk of the organism.
3.6 Complete Reaction Network
4 Simulation Results
The time course of the molecule numbers NB, NC, ND, and NF during linear decay reactions is shown in Figure 2. While NB decays linearly in time, NF grows linearly in time. There are practically no free D molecules during the process, and NC is nearly constant over time.
Figure 3 shows the time course of molecule numbers NB, showing a series of peaks with a PL distribution of heights and widths. Each peak rises very fast and decays linearly with a constant rate, so that the ratio of height to width is the same for all peaks. To demonstrate the scale-invariance of the signal, we also show a zoom into the time course in Figure 4.
All above described results are very stable and highly robust to changes in rate constants. We generated 100 different reaction networks with modified rate constants by adding Gaussian-distributed random numbers with zero mean and standard deviation 0.2 to the reaction rates' default values to test the dependence of the essential functional modules on the parameter settings. In each case the resulting reaction networks yielded very similar results.
So far, we have shown analytically that the peak heights and peak widths of the fluctuating molecule number NB(t) are distributed according to a power law. If this is true, we would expect that the random time series s(t) = NB(t) has long-term correlations. To test this numerically, we have computed the autocorrelation function Css(τ) for different values of κ1 and κ5 (Figure 5). Indeed, we find that Css(τ) decays for long lag times as a PL with a fractional exponent that depends on the parameters κ1 and κ5.
In the recent years, it has been increasingly acknowledged that biochemical reaction networks do not only work as deterministic machines, but that the stochastic nature of reactions, especially at low copy numbers of the molecules, can lead to useful functions for a cell [10, 15]. For example, it has been shown that covalent modification cycles can produce concentration fluctuations with Gaussian, exponential, or PL distributions and with tunable parameters .
In this work, we have demonstrated that also long-time correlated concentration fluctuations of a signaling substance can be generated by a system of relatively simple biochemical reactions. Coupling the concentration fluctuation to the motor units of an organism leads to scale-free random walks, which may be advantageous for efficient foraging and exploration. We therefore hypothesize that the essential functional modules of our reaction network (exponential concentration growth, linear concentration decay) may be identified in signaling pathways of single-cell organisms.
The net balance σnm of the nth species in the mth reaction is defined as σnm = ν′nm − νnm, where νnm ≥ 0 and ν′nm ≥ 0 are the stoichiometry coefficients of the reactants and products, respectively.
Experimental Otolaryngology, ENT-Hospital, Head and Neck Surgery, Friedrich-Alexander University Erlangen-Nuremberg (FAU), Germany. E-mail: email@example.com
Department of Physics, Center for Medical Physics and Technology, Biophysics Group, Friedrich-Alexander University Erlangen-Nuremberg (FAU), Germany. E-mail: firstname.lastname@example.org