Debasis Dan

Solving the advection-diffusion equations in biological contexts using the cellular Potts model

The cellular Potts model (CPM) is a robust, cell-level methodology for simulation of biological tissues and morphogenesis. Both tissue physiology and morphogenesis depend on diffusion of chemical morphogens in the extra-cellular fluid or matrix (ECM). Standard diffusion solvers applied to the cellular potts model use finite difference methods on the underlying CPM lattice. However, these methods produce a diffusing field tied to the underlying lattice, which is inaccurate in many biological situations in which cell or ECM movement causes advection rapid compared to diffusion. Finite difference schemes suffer numerical instabilities solving the resulting advection-diffusion equations. To circumvent these problems we simulate advection diffusion within the framework of the CPM using off-lattice finite-difference methods. We define a set of generalized fluid particles which detach advection and diffusion from the lattice. Diffusion occurs between neighboring fluid particles by local averaging rules which approximate the Laplacian. Directed spin flips in the CPM handle the advective movement of the fluid particles. A constraint on relative velocities in the fluid explicitly accounts for fluid viscosity. We use the CPM to solve various diffusion examples including multiple instantaneous sources, continuous sources, moving sources, and different boundary geometries and conditions to validate our approximation against analytical and established numerical solutions. We also verify the CPM results for Poiseuille flow and Taylor-Aris dispersion.

Mobility and stochastic resonance in spatially inhomogeneous systems.

The mobility of an overdamped particle, in a periodic potential tilted by a constant external field and moving in a medium with periodic friction coefficient is examined. When the potential and the friction coefficient have the same periodicity but have a phase difference, the mobility shows many interesting features as a function of the applied force, the temperature, etc. The mobility shows stochastic resonance even for constant applied force, an issue of much recent interest. The mobility also exhibits a resonance-like phenomenon as a function of the field strength and noise induced slowing down of the particle in an appropriate parameter regime.

Giant diffusion and coherent transport in tilted periodic inhomogeneous systems.

We investigate the dynamics of an overdamped Brownian particle moving in a washboard potential with space dependent friction coefficient. Analytical expressions have been obtained for current and diffusion coefficient. We show that the effective diffusion coefficient can be enhanced or suppressed compared to that of the uniform friction case. The diffusion coefficient is maximum near the critical threshold (F(c)), which is sensitive to temperature and the frictional profile. In some parameter regime, we observe that increase in noise (temperature) decreases the diffusion, which is counterintuitive. This leads to coherent transport with large mean velocity accompanied by small diffusion. This is shown explicitly by analysis of Péclet number, which has been introduced to study coherent or optimal transport. This phenomenon is complementary to giant diffusion.

Multiple current reversals in forced inhomogeneous ratchets

Transport properties of overdamped Brownian particles in a rocked thermal ratchet with space dependent friction coefficient is studied. By tuning the parameters, the direction of current exhibit multiple reversals, both as a function of the thermal noise strength as well as the amplitude of rocking force. Current reversals also occur under deterministic conditions and exhibit intriguing structure. All these results arise due to mutual interplay between potential asymmetry, noise, driving frequency, and inhomogeneous friction.

Motion in a rocked ratchet with spatially periodic friction

We present a detailed study of transport and energetics of a Brownian particle moving in a periodic potential in the presence of an adiabatic external periodic drive. The particle is considered to move in a medium with periodic space dependent friction with the same periodicity as that of the potential but with a phase lag. We obtain several results. Most of the results arise due to the medium being inhomogeneous and are sensitive to the phase lag. When the potential is symmetric we show that the efficiency of energy transduction can be maximized as a function of noise strength or temperature. However, in the case of asymmetric potential the temperature may or may not facilitate the energy conversion. But current reversals can be obtained as a function of temperature and the amplitude of the periodic drive. The reentrant behaviour of the current can also be seen as a function of phase lag.

Stochastic resonance in washboard potentials

We study the mobility of an overdamped particle in a periodic potential tilted by a constant force. The mobility exhibits a stochastic resonance in inhomogeneous systems with space dependent friction coefficient. The result indicates that the presence of an oscillating external field is not essential for the observability of stochastic resonance, at least in an inhomogeneous medium.

EFFICIENCY AND CURRENT REVERSALS IN SPATIALLY INHOMOGENEOUS RATCHETS

Efficiency of generation of net unidirectional current in an adiabatically driven symmetric periodic potential system is studied. The efficiency shows a maximum, in the case of an inhomogeneous system with spatially varying periodic friction coefficient, as a function of temperature. The ratchet is not most efficient when it gives maximum current. The direction of current may also be reversed as a function of noise strength when, instead, an asymmetric periodic potential is considered.

Energetics of rocked inhomogeneous ratchets

We study the efficiency of frictional thermal ratchets driven by a finite frequency driving force and in contact with a heat bath. The efficiency exhibits varied behavior with driving frequency. Both nonmonotonic and monotonic behavior have been observed. In particular, the magnitude of the efficiency in the finite frequency regime may be more than the efficiency in the adiabatic regime. This is our central result for rocked ratchets. We also show that for the simple potential we have chosen, with only spatial asymmetry (homogeneous system) or only a frictional ratchet (symmetric potential profile), the adiabatic efficiency is always more than in the nonadiabatic case.

COHERENCE IN TRANSPORT IN A SPECIAL CLASS OF INHOMOGENEOUS RATCHET

We study the noise induced transport of an overdamped Brownian particle in frictional ratchet systems in the presence of a special class of external multiplicative Gaussian white noise fluctuations. The analytical expressions for current and diffusion coefficient are derived and the reliability or coherence of transport are discussed by means of their ratio. We show that these frictional ratchets exhibit larger coherence as compared to the flashing and rocking ratchets.