Sergei Fedotov

Migration and proliferation dichotomy in tumor cell invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW), we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting-time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.

Flamelet modelling of non-premixed turbulent combustion with local extinction and re-ignition

Extinction and re-ignition in non-premixed turbulent combustion is investigated. A flamelet formulation accounting for transport along mixture fraction iso-surfaces is developed. A new transport term appears in the flamelet equations, which is modelled by a stochastic mixing approach. The timescale appearing in this model is obtained from the assumption that transport at constant mixture fraction is only caused by changes of the local scalar dissipation rate. The space coordinates appearing in this term can then be replaced by the mixture fraction and the scalar dissipation rate. The dissipation rate of the scalar dissipation rate appears as a diffusion coefficient in the new term. This is a new parameter of the problem and is called the re-ignition parameter. The resulting equations are simplified and stochastic differential equations for the scalar dissipation rate and the re-ignition parameter are formulated. The system of equations is solved using Monte Carlo calculations. The results show that the newly appearing transport term acts by modifying the S-shaped curve such that the lower turning point appears at higher scalar dissipation rate. In an a priori study, predictions using this model are compared with data from a direct numerical simulation of non-premixed combustion in isotropic turbulence simulating extinction and re-ignition.

Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts

The main aim of the paper is to incorporate the nonlinear kinetic term into non-Markovian transport equations described by a continuous time random walk (CTRW) with nonexponential waiting time distributions. We consider three different CTRW models with reactions. We derive nonlinear Master equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems with Kolmogorov-Petrovskii-Piskunov kinetics and anomalous diffusion. We have found an explicit expression for the speed of a propagating front in the case of subdiffusive transport.

Investigation of scalar dissipation rate fluctuations in non-premixed turbulent combustion using a stochastic approach

Turbulent fluctuations of the scalar dissipation rate are well known to have a strong impact on ignition and extinction in non-premixed combustion. In the present study the influence of stochastic fluctuations of the scalar dissipation rate on the solution of the flamelet equations is investigated. A one-step irreversible reaction is assumed. The system can thereby be described by the solution of the temperature equation. By modelling the diffusion terms in the flamelet equation this can be written as an ordinary stochastic differential equation (SDE). In addition, an SDE is derived for the scalar dissipation rate. From these two equations, a Fokker-Planck equation can be obtained describing the joint probability of temperature and the scalar dissipation rate. The equation is analysed and integrated numerically using a fourth-order Runge-Kutta scheme. The influence of the main parameters, which are the Damköhler number, the Zeldovich number, the heat release parameter and the variance of the scalar dissipation rate fluctuations, are discussed.

OPTION PRICING FOR INCOMPLETE MARKETS VIA STOCHASTIC OPTIMIZATION: TRANSACTION COSTS, ADAPTIVE CONTROL AND FORECAST

The problem of determining the European-style option price in incomplete markets is examined within the framework of stochastic optimization. An analytic method based on the stochastic optimization is developed that gives the general formalism for determining the option price and the optimal trading strategy (optimal feedback control) that reduces the total risk inherent in writing the option. The cases involving transaction costs, the stochastic volatility with uncertainty, stochastic adaptive process, and forecasting process are considered. A software package for the option pricing for incomplete markets is developed and the results of numerical simulations are presented.

Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion

The proliferation and migration dichotomy of the tumor cell invasion is examined within a two-component continuous time random walk (CTRW) model. The balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived. The transport of tumor cells is formulated in terms of the CTRW with an arbitrary waiting time distribution law, while proliferation is modeled by a logistic growth. The overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined.

Nonlinear degradation-enhanced transport of morphogens performing subdiffusion.

We study a morphogen gradient formation under nonlinear degradation and subdiffusive transport. In the long-time limit, we obtain the nonlinear effect of degradation-enhanced diffusion, resulting from the interaction of non-Markovian subdiffusive transport with a nonlinear reaction. We find the stationary profile of power-law type, which has implications for robustness, with the shape of the profile being controlled by the anomalous exponent. Far away from the source of morphogens, any changes in the rate of production are not felt.

Density-dependent dispersal and population aggregation patterns

We have derived reaction-dispersal-aggregation equations from Markovian reaction-random walks with density-dependent jump rate or density-dependent dispersal kernels. From the corresponding diffusion limit we recover well-known reaction-diffusion-aggregation and reaction-diffusion-advection-aggregation equations. It is found that the ratio between the reaction and jump rates controls the onset of spatial patterns. We have analyzed the qualitative properties of the emerging spatial patterns. We have compared the conditions for the possibility of spatial instabilities for reaction-dispersal and reaction-diffusion processes with aggregation and have found that dispersal process is more stabilizing than diffusion. We have obtained a general threshold value for dispersal stability and have analyzed specific examples of biological interest.

Anomalous transport and nonlinear reactions in spiny dendrites

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particles position increases as t(μ) with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

G-equation, stochastic control theory and relativistic mechanics of a particle moving in a random field

The problem of turbulent premixed flames is considered within the framework of the field equations describing flames as level surfaces for scalar fields and stochastic control theory for Markov diffusion processes. It is shown that all field equations can be interpreted as a dynamic programming partial differential equation of second order (Hamilton - Jacobi - Bellman PDE) and the corresponding scalar fields can be regarded as the value functions. The explicit formulae for the scalar fields as a minimum/maximum of a functional integral have been derived and the most interesting result is that the running cost function is the same as a Lagrangian function for a relativistic particle moving in an external field and the particle velocity plays the role of a control function.