In recent years, there has been increased awareness that stochasticity in chemical reactions and diffusion of molecules can have significant effects on the outcomes of intracellular processes, particularly given the low copy numbers of many proteins and mRNAs present in a cell. For such molecular species, the number and locations of molecules can provide a more accurate and detailed description than local concentration. In addition to diffusion, drift in the movements of molecules can play a key role in the dynamics of intracellular processes, and can often be modeled as arising from potential fields. Examples of sources of drift include active transport, variations in chemical potential, material heterogeneities in the cytoplasm, and local interactions with subcellular structures. This dissertation presents a new numerical method for simulating the stochastically varying numbers and locations of molecular species undergoing chemical reactions and drift-diffusion. The method combines elements of the First-Passage Kinetic Monte Carlo (FPKMC) method for reaction-diffusion systems and the Wang—Peskin—Elston lattice discretization of the Fokker—Planck equation that describes drift-diffusion processes in which the drift arises from potential fields. In the FPKMC method, each molecule is enclosed within a "protective domain," either by itself or with a small number of other molecules. To sample when a molecule leaves its protective domain or a reaction occurs, the original FPKMC method relies on analytic solutions of one- and two-body diffusion equations within the protective domains, and therefore cannot be used in situations with non-constant drift. To allow for such drift in our new method (hereafter Dynamic Lattice FPKMC or DL-FPKMC), each molecule undergoes a continuous-time random walk on a lattice within its protective domain, and the lattices change adaptively over time. One of the most commonly used spatial models for stochastic reaction-diffusion systems is the Smoluchowski diffusion-limited reaction (SDLR) model. The DL-FPKMC method generates convergent realizations of an extension of the SDLR model that includes drift from potentials. We present detailed numerical results demonstrating the convergence and accuracy of our method for various types of potentials (smooth, discontinuous, and constant). We also present several illustrative applications of DL-FPKMC, including examples motivated by cell biology.

Parameter inference for stochastic systems is considered as one of the fundamental classical problems in the domain of computational systems biology. The problem becomes challenging and often analytically intractable with the large number of uncertain parameters. In this scenario, Markov Chain Monte Carlo (MCMC) algorithms have been proved to be highly effective. For a stochastic system, the most accurate description of the kinetics is given by the Chemical Master Equation (CME). Unfortunately, analytical solution of CME is often intractable even for considerably small amount of chemically reacting species due to its super exponential state space complexity. As a solution, Stochastic Simulation Algorithm (SSA) using Monte Carlo approach was introduced to simulate the chemical process defined by the CME. SSA is an exact stochastic method to simulate CME but it also suffers from high time complexity due to simulation of every reaction. Therefore computation of likelihood function (based on exact CME) in MCMC becomes expensive which alternately makes the rejection step expensive. In this generic work, we introduce different approximations of CME as a pre-conditioning step to the full MCMC to make rejection cheaper. The goal is to avoid expensive computation of exact CME as far as possible. We show that, with effective pre-conditioning scheme, one can save a considerable amount of exact CME computations maintaining similar convergence characteristics. Additionally, we investigate three different sampling schemes (dense sampling, longer sampling and i.i.d sampling) under which convergence for MCMC using exact CME for parameter estimation can be analyzed. We find that under i.i.d sampling scheme, better convergence can be achieved than that of dense sampling of the same process or sampling the same process for longer time. We verify our theoretical findings for two different processes: linear birth-death and dimerization.Apart from providing a framework for parameter inference using CME, this work also provides us the reasons behind avoiding CME (in general) as a parameter estimation technique for so long years after its formulation

An object-oriented framework is presented for developing and simulating individual-based models of cell populations. The framework supplies classes to define objects called simulation channels that encapsulate the algorithms that make up a simulation model. These may govern state-updating events at the individual level, perform global state changes, or trigger cell division. Simulation engines control the scheduling and execution of collections of simulation channels, while a simulation manager coordinates the engines according to one of two scheduling protocols. When the ensemble of cells being simulated reaches a specified maximum size, a procedure is introduced whereby random cells are ejected from the simulation and replaced by newborn cells to keep the sample population size constant but representative in composition. The framework permits recording of population snapshot data and/or cell lineage histories. Use of the framework is demonstrated through validation benchmarks and two case studies based on experiments from the literature.

An object-oriented framework is presented for developing and simulating individual-based models of cell populations. The framework supplies classes to define objects called simulation channels that encapsulate the algorithms that make up a simulation model. These may govern state-updating events at the individual level, perform global state changes, or trigger cell division. Simulation engines control the scheduling and execution of collections of simulation channels, while a simulation manager coordinates the engines according to one of two scheduling protocols. When the ensemble of cells being simulated reaches a specified maximum size, a procedure is introduced whereby random cells are ejected from the simulation and replaced by newborn cells to keep the sample population size constant but representative in composition. The framework permits recording of population snapshot data and/or cell lineage histories. Use of the framework is demonstrated through validation benchmarks and two case studies based on experiments from the literature.

An object-oriented framework is presented for developing and simulating individual-based models of cell populations. The framework supplies classes to define objects called simulation channels that encapsulate the algorithms that make up a simulation model. These may govern state-updating events at the individual level, perform global state changes, or trigger cell division. Simulation engines control the scheduling and execution of collections of simulation channels, while a simulation manager coordinates the engines according to one of two scheduling protocols. When the ensemble of cells being simulated reaches a specified maximum size, a procedure is introduced whereby random cells are ejected from the simulation and replaced by newborn cells to keep the sample population size constant but representative in composition. The framework permits recording of population snapshot data and/or cell lineage histories. Use of the framework is demonstrated through validation benchmarks and two case studies based on experiments from the literature.

Chemical reactions involving small number of reacting molecules are noisy processes. They are simulated using stochastic simulation algorithms like the Gillespie SSA, which are valid when the reaction environment is well-mixed. This is not the case in reactions occuring on biological media like cell membranes, where alternative simulation methods have to be used to account for the crowded nature of the reacting environment. Ion channels, which are membrane proteins controlling the flow of ions into and out of the cell, offer excellent single molecule conditions to test stochastic simulation schemes in crowded biological media. Single molecule reactions are of great importance in determining the functions of biological molecules. Access to their experimental data have increased the scope of com-putational modeling of biological processes. Recently, single molecule experiments have revealed the non-Markovian nature of chemical reactions, due to a phenomenon called `dynamic disorder', which makes the rate constants a deterministic function of time or a random process. This happens when there are additional slow scale conformational transitions, giving the molecule a memory of its previous states. In a previous work, the hTREK1 two pore domain potassium channel was revealed to have long term memory in its kinetics, prompting alternate non-Markovian schemes to analyze its gating. Traditionally, ion channel gating is modeled as Markovian transitions between fixed states. In this work, we have used single channel data from hTREK1 ion channel and have provided a simple diffusion model for its gating. The main assumption of this model is that the ion channel diffuses through a continuum of states on its potential energy landscape, which is derived from the steady state probability distribution of ionic current recorded from patch clamp experiments. A stochastic differential equation (SDE) driven by Gaussian white noise is proposed to model this motion in an asymmetric double well potential. The method is computationally very simple and efficient and reproduces the amplitude histogram very well. For the case when ligands are added, leading to incorporation of long term memory in the kinetics, the SDE is modified to run on coloured noise. This has been done by introducing an auxiliary variable into the equation. It has been shown that increasing the noise correlation with ligand concentration improves the fits to the experimental data. This has been validated for several datasets. These methods are more advantageous for simulation than the Markovian models as they are true to the physical picture of gating and also computationally very efficient. Reproducing the whole raw data trace takes no more than a few seconds with our scheme, with the only input being the amplitude histogram and four parameters. Finally a quantitative model based on a modified version of the Chemical Langevin equation is given, which works on random rate parameters. This model is computationally simple to implement and reproduces the catalytic activity of the channel as a function of time. From the computational analysis undertaken in this work, we can infer that ion channel activity can be modeled using the framework of non-Markovian processes, lending credence to the recent understanding that single molecule reactions are basically processes with long-term memory. Since the ion channel is basically a protein, we can also hypothesize that the some of the properties that make proteins so vital to living organ-isms could be attributed to long-term memory in their folding kinetics, giving them the ability to sample specific regions of their conformation space, which are of interest to biological functions.

An object-oriented framework is presented for developing and simulating individual-based models of cell populations. The framework supplies classes to define objects called simulation channels that encapsulate the algorithms that make up a simulation model. These may govern state-updating events at the individual level, perform global state changes, or trigger cell division. Simulation engines control the scheduling and execution of collections of simulation channels, while a simulation manager coordinates the engines according to one of two scheduling protocols. When the ensemble of cells being simulated reaches a specified maximum size, a procedure is introduced whereby random cells are ejected from the simulation and replaced by newborn cells to keep the sample population size constant but representative in composition. The framework permits recording of population snapshot data and/or cell lineage histories. Use of the framework is demonstrated through validation benchmarks and two case studies based on experiments from the literature.