Living organisms contain thousands of interacting complex networks of macromolecules, and there are very few tools to understand them. The present thesis is an attempt to elucidate some aspects of two interacting networks which are responsible for the coordinated chemotactic locomotion of a cell. One of the networks, the directional sensing network, is responsible for the reception of external molecular signals to which the cell is exposed. This network also transforms the external signal into an internal one (`response'), which is amplified and used by the the second network, the polarization network, to tune and to guide the cellular motor which propells the cell using its polymerizing cytoskeleton. A stochastic model, a type of cellular automata model, has been developed and employed in order to address various questions. For example, how an external signal with a weak spatial gradient can be translated by molecules into a strongly amplified and localized response, and how this response regulates the local activity and the spatial distribution of the actin cytoskeleton which controls the velocity and the direction of the cell's movements. By using a stochastic model, which includes explicit particles, the investigations provide a link to known approaches in theoretical physics, as there are, e.g., cooperative phenomena in many-body problems and space-time correlations in nonlinear dynamics. Since the present study is the first attempt employing a stochastic model, as compared to previous kinetic and deterministic models for chemotaxis, the achieved results contain new and important information. It is shown, among others, that the amplification of the response exhibits a transition as function of the gradient of the signal. The spatial localization of the response, represented by the distribution of activated PIP molecules along the cell membrane, depends on the gradient and the maximum of the signal. Using the `Local Exciter and Global Inhibitor' (LEGI) model, proposed recently by other researchers for the directional sensing network, it is shown how the spacial-temporal distributions of the two types of inhibitor and exciter molecules are correlated to the amplification of the response in terms of activated PIP molecules. The major advantage of the present approach, however, is the combination of a particle-based LEGI network with a particle-based polarization network, where the latter includes explicitly linear and branching polymerization of actin filaments. Taking the two regulatory networks, including their signaling molecules and the actin molecules together, a minimal cell model has been developed, where the cell membrane is represented by a two-dimensional flexible ring polymer. During Monte Carlo simulations of this model, the chemotactic motion of the cell could be monitored. The analysis of the trajectories shows that the magnitude of the drift velocity can be tuned by the combination of the signal gradient, the signal maximum and the signal-mediated polymerization of the filaments. This explains the experimentally known high sensitivity of chemotactic cell to weak external signal gradients.