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Mathematical and Physical Foundations
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Introduction
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Daphne is a software system for creating and running the simulations of multi-cellular systems.  This document provides guidance on the mathematics behind the simulation system.

Daphne is built on the generic processes of Newtonian mechanics, and generalized chemical kinetics, as well as a small number of cell biology-specific constructs. That is, for example, any entity that has a position changes its position in accordance with Newton’s laws. Forces arise in the mechanical interactions of cells and as a result of molecular interactions and molecular conformational changes. Collections of entities, typically cells or molecules, can be treated as continua and described by their concentration or density fields as well as by a lists of individual positions (and velocities). These fields obey kinetic equations derived from mass-action kinetics.

Molecular concentrations and related quantities are often modeled as continuous scalar fields in space that evolve continuously in time in response to physical processes, such as reaction and diffusion.  Obtaining analytical solutions for these quantities at all spatial and time points is generally not possible for complex biological systems, and numerical techniques must be used to obtain approximations of the dynamical behavior of the system.  Generally, techniques for approximating the temporal and spatial aspects of the problem are implemented independently.

Currently in Daphne, we implement two types of approximations for the spatial dependence of the system, moment-expansion and discrete, each with a distinct scalar field representation.  In the moment-expansion scalar field representation, the spatial dependence of a quantity is represented by an infinite series expansion, with each successive term accounting for the next (higher-order) moment (of the spatial distribution) of the system.  The spatial accuracy of the moment-expansion approximation is determined by the number of terms in the infinite series that are retained for calculations.  With this numerical approach, the quantities of interest are still functions of continuous spatial variables.  Currently, Daphne utilizes two-term moment-expansion fields to describe scalar fields in cells.  The advantage of this representation is that it provides computational efficiency while still providing information about the spatial distribution, or polarity, of a scalar field, through the second (dipole) term.  This polarity is crucial for modeling cellular processes, such as chemotaxis.

In contrast, the discretized numerical approach seeks solutions to the dynamical equations at only a (finite) set of discrete spatial points in the space of interest, and the values of quantities at intermediate spatial points are approximated through interpolation of the solutions at the discrete (lattice) points.  The spatial accuracy of the discretized approximation is determined by the density of discrete points and the size of the neighborhood (number of nearest neighbors) that is used to approximate the equations.

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