Stall force of a cargo driven by N interacting motor proteins

We study a generic one-dimensional model for an intracellular cargo driven by N motor proteins against an external applied force. The model includes motor-cargo and motor-motor interactions. The cargo motion is described by an over-damped Langevin equation, while motor dynamics is specified by hopping rates which follow a local detailed balance condition with respect to change in energy per hopping event. Based on this model, we show that the stall force, the mean external force corresponding to zero mean cargo velocity, is completely independent of the details of the interactions and is, therefore, always equal to the sum of the stall forces of the individual motors. This exact result is arrived on the basis of a simple assumption: the (macroscopic) state of stall of the cargo is analogous to a state of thermodynamic equilibrium, and is characterized by vanishing net probability current between any two microstates, with the latter specified by motor positions relative to the cargo. The corresponding probability distribution of the microstates under stall is also determined. These predictions are in complete agreement with numerical simulations, carried out using specific forms of interaction potentials.Comment: Accepted in Europhysics Letter

Transport of organelles by elastically coupled motor proteins

Motor-driven cargo transport is a complex phenomenon where multiple motor proteins attached on to a cargo engage in pulling activity, often leading to tug-of-war, displaying bidirectional motion. However, most mathematical and computational models ignore the details of the motor-cargo interaction. Here, we study a generic model in which N motors are elastically coupled to a cargo, which itself is subjected to thermal noise in the cytoplasm and to an additional external applied force. The motor-hopping rates are chosen to satisfy detailed balance with respect to the energy of elastic stretching. With these assumptions, an (N+1)-variable master equation is constructed for dynamics of the motor-cargo complex. By expanding the hopping rates to linear order in fluctuations in motor positions, we obtain a linear Fokker-Planck equation. The deterministic equations governing the average quantities are separated out and explicit analytical expressions are obtained for the mean velocity and diffusion coefficient of the cargo. We also study the statistical features of the force experienced by an individual motor and quantitatively characterize the load-sharing among the cargo-bound motors. The mean cargo velocity and the effective diffusion coefficient are found to be decreasing functions of the stiffness. While increase in the number of motors N does not increase the velocity substantially, it decreases the effective diffusion coefficient which falls as 1/N asymptotically. We further show that the cargo-bound motors share the force exerted on the cargo equally only in the limit of vanishing elastic stiffness; as stiffness is increased, deviations from equal load sharing are observed. Numerical simulations agree with our analytical results where expected. Interestingly, we find in simulations that the stall force of a cargo elastically coupled to motors is independent of the stiffness

Ultrasensitivity and Fluctuations in the Barkai-Leibler Model of Chemotaxis Receptors in {\it Escherichia coli}

A stochastic version of the Barkai-Leibler model of chemotaxis receptors in {\it E. coli} is studied here to elucidate the effects of intrinsic network noise in their conformational dynamics. It was originally proposed to explain the robust and near-perfect adaptation of {\it E. coli} observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. A receptor is either active or inactive and can stochastically switch between the two states. Enzyme CheR methylates inactive receptors while CheB demethylates active ones and the probability for it to be active depends on its level of methylation and ligandation. A simple version of the model with two methylation sites per receptor (M=2) shows zero-order ultrasensitivity (ZOU) akin to the classical 2-state model of covalent modification studied by Goldbeter and Koshland (GK). For extremely small and large ligand concentrations, the system reduces to two 2-state GK modules. A quantitative measure of the spontaneous fluctuations in activity (variance) estimated mathematically under linear noise approximation (LNA) is found to peak near the ZOU transition. The variance is a weak, non-monotonic and decreasing functions of ligand and receptor concentrations. Gillespie simulations for M=2 show excellent agreement with analytical results obtained under LNA. Numerical results for M=2, 3 and 4 show ZOU in mean activity; the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to have a beneficial effect on the motility of the organism. The response of mean receptor activity to small time-dependent changes in the external ligand concentration, computed within linear response theory, is found to have a bilobe form.Comment: Accepted in PLoS On

Bond percolation of polymers

We study bond percolation of $N$ non-interacting Gaussian polymers of $\ell$ segments on a 2D square lattice of size $L$ with reflecting boundaries. Through simulations, we find the fraction of configurations displaying {\em no} connected cluster which span from one edge to the opposite edge. From this fraction, we define a critical segment density $\rho_{c}^L(\ell)$ and the associated critical fraction of occupied bonds $p_{c}^L(\ell)$, so that they can be identified as the percolation threshold in the $L \to \infty$ limit. Whereas $p_{c}^L(\ell)$ is found to decrease monotonically with $\ell$ for a wide range of polymer lengths, $\rho_{c}^L(\ell)$ is non-monotonic. We give physical arguments for this intriguing behavior in terms of the competing effects of multiple bond occupancies and polymerization.Comment: 4 pages with 6 figure

Stationary states of an active Brownian particle in a harmonic trap

We study the stationary states of an over-damped active Brownian particle (ABP) in a harmonic trap in two dimensions, via mathematical calculations and numerical simulations. In addition to translational diffusion, the ABP self-propels with a certain velocity, whose magnitude is constant, but its direction is subject to Brownian rotation. In the limit where translational diffusion is negligible, the stationary distribution of the particle's position shows a transition between two different shapes, one with maximum and the other with minimum density at the centre, as the trap stiffness is increased. We show that this non-intuitive behaviour is captured by the relevant Fokker-Planck equation, which, under minimal assumptions, predicts a continuous ``phase transition" between the two different shapes. As the translational diffusion coefficient is increased, both these distributions converge into the equilibrium, Boltzmann form. Our simulations support the analytical predictions, and also show that the probability distribution of the orientation angle of the self-propulsion velocity undergoes a transition from unimodal to bimodal forms in this limit. We also extend our simulations to a three dimensional trap, and find similar behaviour