Differential equations frameworks and models for the physics of biological systems

Preface

In the last and in the present centuries the interest towards the biological systems has largely increased considering the recent progression in the natural sciences. Usually, the analysis of a biological system is expensive from the in vivo and in vitro experiments viewpoint. Indeed, in the case of an in vivo experiment the use of animals entails both ethical and availability issues [1]; in the case of an in vitro experiment, the results can be misleading considering that the normal biological environment is limited [2]. A summary of the advantages and disadvantages of in vivo and in vitro methods can be found in [3].

Biological systems are composed by elements that, differently from the inert matter, can perform a strategy or a function. The interactions among the constituting elements occur at different levels and sometimes it is not necessary the contact because (cellular) signals are usually at the base of interactions [4]. The result of interactions can be also proliferation/destruction or mutation of the elements. Accordingly, the classical laws of physics can be broken. Moreover, biological systems are characterized by emerging collective behaviors that are not the simple linear combination of the interactions. The microscopic state of a component of a biological system thus is composed by the classical mechanical variables, e.g. space and velocity, but also by an internal-state variable, which models the strategy/function. According to the above-mentioned properties, a biological system has thus the structure of a complex system [5]–[7].

The analysis of a complex biological system requires also interactions among different research fields. A multidisciplinary approach needs to be employed and the development of such an approach demands the definition of specific frameworks and models.

The most developed approaches come from mathematics, physics and information sciences. It is worth stressing that the notion of model in the previous mentioned research fields is different from the classical notion of model in biology [8]. Indeed, the term model is employed in biology to identify a non-human species subjected to some experiments. Differently from biology, the term model in mathematics or physics represents a system of equations that describe the time evolution of a quantity related to the system under consideration.

As already mentioned, the collaboration between different scholars is an important issue in the modeling of a complex biological system. Recently three main multidisciplinary research domains have emerged: biomathematics [9], biophysics-biomechanics [10],[11] and bioinformatics [12].

In biomathematics and biophysics, the term differential equation framework defines a differential equation, or a system of differential equations fulfilled by some quantities related to the system. A differential equation is an identity relating more functions and their derivatives. Each differential equation is defined by the introduction of some coefficients and functions.

A specific differential equation model is obtained once these parameters and functions are quantitatively defined and all the initial or boundary conditions are defined. Accordingly, a differential equation model is defined by recurring to a Cauchy and/or to a Dirichlet-Neumann-Robin problem, usually called initial-value and initial-boundary-value problems, respectively. Finally, a differential equation model is well-posed in the Hadamard sense if the solution exists, it is unique and depends continuously on the initial data.

Three main differential equation frameworks can be found in the pertinent literature: The ordinary differential equation (ODE) framework, the partial differential equation (PDE) framework and the integro-differential equation (IDE) framework. In the ODE-framework only the time evolution of the quantities is modeled whereas in the PDE-framework also the role of the mechanical and internal-state variables can be taken into consideration. An IDE-framework, which is defined by introducing both integrals and derivatives, can model either the time evolution or the time-space-velocity-internal-state evolution of the quantities under consideration.

In this century two new developments have been proposed:

Bearing all above in mind, different approaches have been derived in the context of differential equation frameworks. Among others:

It is worth stressing that various mathematical theories have been involved in the modeling of complex biological systems, among others, the inverse theory, the non-equilibrium statistical mechanics theory, the information theory, the game theory, the asymptotic theory.

An important issue that needs to be considered when dealing with the complex biological systems is the scale problem. Indeed, each phenomenon can be analyzed at a different scale. Usually, three main representation scales have been identified: microscale, mesoscale and macroscale [18].

The following important question raises:

What is the most suitable differential equation framework at a specific scale?

The answer is not obvious considering that each framework can present advantages and disadvantage from the computational viewpoint and that each framework can require assumptions that can reduce the analysis of the biological phenomena. Indeed, considering the complexity of the biological system, a phenomenological analysis is necessary to identify the main actors, interactions and in particular to reduce the number of parameters of the model.

The most accredited classification establishes that usually phenomena occurring at:

It is worth pointing out that the differential equation framework is not the only modeling structure presented in the literature. Indeed, bioinformatics [19] has proposed further modeling approaches based on the so-called agent-based model that can be considered as a microscopic approach based on the definition of agent which is a discrete entity capable to perform a strategy. An agent modifies its behaviors regulated by specific rules defined in the model [20]. In this context, graph theory, cellular automata, lattice Boltzmann models have been also defined.

Is it worth stressing that even if the models of bioinformatics are not defined by differential equations, they can be used to tune some parameters/function of a differential equation model thus constructing a hybrid model.

As already mentioned, a phenomenon at a specific scale can be modeled with a specific differential equation framework. However, each phenomenon obviously depends on other phenomena occurring at different scales. Accordingly, a multi-scale approach is necessary for linking the different interactions presented in the different scales. This is the most important issue that can be pursued by considering the interplay among the different scholars coming from the different research domains. Even if the interest in this research domain has increased, the results are not satisfactory and further developments are required.

Finally, a differential equation model needs to be validated with empirical or experimental data. The latter step demands a further interplay with scholars coming from the bioinformatics domain, see, among others, the paper [21].