Random intermittent search and the tug-of-war model of motor-driven transport

We formulate the tug-of-war model of microtubule cargo transport by multiple molecular motors as an intermittent random search for a hidden target. A motor-complex consisting of multiple molecular motors with opposing directional preference is modeled using a discrete Markov process. The motors randomly pull each other off of the microtubule so that the state of the motor-complex is determined by the number of bound motors. The tug-of-war model prescribes the state transition rates and corresponding cargo velocities in terms of experimentally measured physical parameters. We add space to the resulting Chapman-Kolmogorov (CK) equation so that we can consider delivery of the cargo to a hidden target somewhere on the microtubule track. Using a quasi-steady state (QSS) reduction technique we calculate analytical approximations of the mean first passage time (MFPT) to find the target. We show that there exists an optimal adenosine triphosphate (ATP)concentration that minimizes the MFPT for two different cases: (i) the motor complex is composed of equal numbers of kinesin motors bound to two different microtubules (symmetric tug-of-war model), and (ii) the motor complex is composed of different numbers of kinesin and dynein motors bound to a single microtubule(asymmetric tug-of-war model)

Filling of a Poisson trap by a population of random intermittent searchers

We extend the continuum theory of random intermittent search processes to the case of $N$ independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi--infinite track. Each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state. We assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process. The hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target; the particle is removed from the system after delivering its cargo. As a further generalization of previous models, we assume that up to $n$ successive particles can find the target and deliver its cargo. Assuming that the rate of target detection scales as $1/N$, we show that there exists a well--defined mean field limit $N\rightarrow \infty$, in which the stochastic model reduces to a deterministic system of linear reaction--hyperbolic equations for the concentrations of particles in each of the internal states. These equations decouple from the stochastic process associated with filling the target with cargo. The latter can be modeled as a Poisson process in which the time--dependent rate of filling $\lambda(t)$ depends on the concentration of stationary particles within the target domain. Hence, we refer to the target as a Poisson trap. We analyze the efficiency of filling the Poisson trap with $n$ particles in terms of the waiting time density $f_n(t)$. The latter is determined by the integrated Poisson rate $\mu(t)=\int_0^t\lambda(s)ds$, which in turn depends on the solution to the reaction-hyperbolic equations. We obtain an approximate solution for the particle concentrations by reducing the system of reaction-hyperbolic equations to a scalar advection--diffusion equation using a quasi-steady-state analysis. We compare our analytical results for the mean--field model with Monte-Carlo simulations for finite $N$. We thus determine how the mean first passage time (MFPT) for filling the target depends on $N$ and $n$

A perturbation analysis of spontaneous action potential initiation by stochastic ion channels

A stochastic interpretation of spontaneous action potential initiation is developed for the Morris- Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently-developed quasi-stationary perturbation method. Using the fact that the activating ionic channel’s random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT

Local synaptic signaling enhances the stochastic transport of\ud motor-driven cargo in neurons

The tug-of-war model of motor-driven cargo transport is formulated as an intermittent trapping process. An immobile trap, representing the cellular machinery that sequesters a motor-driven cargo for eventual use, is located somewhere within a microtubule track. A particle representing a motor-driven cargo that moves randomly with a forward bias is introduced at the beginning of the track. The particle switches randomly between a fast moving phase and a slow moving phase. When in the slow moving phase, the particle can be captured by the trap. To account for the possibility the particle avoids the trap, an absorbing boundary is placed at the end of the track. Two local signaling mechanisms—intended to improve the chances of capturing the target—are considered by allowing the trap to affect the tug-of-war parameters within a small region around itself. The first is based on a localized adenosine triphosphate (ATP) concentration gradient surrounding a synapse, and the second is based on a concentration of tau—a microtubule-associated protein involved in Alzheimer’s disease—coating the microtubule near the synapse. It is shown that both mechanisms can lead to dramatic improvements in the capture probability, with a minimal increase in the mean capture time. The analysis also shows that tau can cause a cargo to undergo random oscillations, which could explain some experimental observations

Stochastic models of intracellular transport

The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an over-damped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of ATP hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review we present a wide range of analytical methods and models of intracellular transport. In the case of diffusive transport, we consider narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion. In the case of active transport, we consider Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean field approximations. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self–organization of subcellular structures

Quasi-steady state reduction of molecular motor-based models of directed intermittent search

We present a quasi–steady state reduction of a linear reaction–hyperbolic master equation describing the directed intermittent search for a hidden target by a motor–driven particle moving on a one–dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile, non-search phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor–driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker–Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V , diffusivity D and target absorption rate λ of the random search. The quasi–steady state reduction determines V, D and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3–state model and show that our quasi–steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a “tug-of-war,” and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets

Effects of demographic noise on the synchronization of a metapopulation in a fluctuating environment

We use the theory of noise-induced phase synchronization to analyze the effects of demographic noise on the synchronization of a metapopulation of predator-prey systems within a fluctuating environment (Moran effect). Treating each local predator–prey population as a stochastic urn model, we derive a Langevin equation for the stochastic dynamics of the metapopulation. Assuming each local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive the steady state probability density for pairwise phase differences between oscillators, which is then used to determine the degree of synchronization of\ud the metapopulation

Correspondence between John Murphy and P. H. Newby: Part 2

Letter from the 1969 prize winner agreeing to offer signed copie

Directed intermittent search for hidden targets

We develop and analyze a stochastic model of directed intermittent search for a hidden target on a one-dimensional track. A particle injected at one end of the track randomly switches between a stationary search phase and a mobile, non-search phase that is biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track or due to competition with other targets. Such a scenario is exemplified by the motor-driven transport of mRNA granules to synaptic targets along a dendrite. We first calculate the hitting probability and conditional mean first passage time (MFPT) for finding a single target. We show that an optimal search strategy does not exist, although for a fixed hitting probability, a unidirectional rather than a partially biased search strategy generates a smaller MFPT. We then extend our analysis to the case of multiple targets, and determine how the hitting probability and MFPT depend on the number of targets

Which outcome expectancies are important in determining young adults intentions to use condoms with casual sexual partners?: A cross-sectional study

Background: The prevalence of unwanted pregnancy and sexually transmitted infection amongst young adults represents an important public health problem in the UK. Individuals attitude towards the use of condoms has been identified as an important determinant of behavioural intentions and action. The Theory of Planned Behaviour has been widely used to explain and predict health behaviour. This posits that the degree to which an individual positively or negatively values a behaviour (termed direct attitude) is based upon consideration of the likelihood of a number of outcomes occurring (outcome expectancy) weighted by the perceived desirability of those outcomes (outcome evaluation). Outcome expectancy and outcome evaluation when multiplied form indirect attitude. The study aimed to assess whether positive outcome expectancies of unprotected sex were more important for young adults with lower safe sex intentions, than those with safer sex intentions, and to isolate optimal outcomes for targeting through health promotion campaigns. Methods: A cross-sectional survey design was used. Data was collected from 1051 school and university students aged 16-24 years. Measures of intention, direct attitude and indirect attitude were taken. Participants were asked to select outcome expectancies which were most important in determining whether they would use condoms with casual sexual partners. Results: People with lower safe sex intentions were more likely than those with safer sex intentions to select all positive outcome expectancies for unprotected sex as salient, and less likely to select all negative outcome expectancies as salient. Outcome expectancies for which the greatest proportion of participants in the less safe sex group held an unfavourable position were: showing that I am a caring person, making sexual experiences less enjoyable, and protecting against pregnancy. Conclusions: The findings point to ways in which the attitudes of those with less safe sex intentions could be altered in order to motivate positive behavioural change. They suggest that it would be advantageous to highlight the potential for condom use to demonstrate a caring attitude, to challenge the potential for protected sex to reduce sexual pleasure, and to target young adults risk appraisals for pregnancy as a consequence of unprotected sex with casual sexual partners