On the mathematical theory of living systems II: The interplay between mathematics and system biology

This paper aims at showing how the so-called mathematical kinetic theory for active particles can be properly developed to propose a new system biology approach. The investigation begins with an analysis of complexity in biological systems, continues with reviewing a general methodology to reduce complexity and furnishes the mathematical tools to describe the time evolution of such systems by capturing all their features. © 2011 Elsevier Ltd. All rights reserved. 1. Aims and plan of the paper The emerging area of system biology aims at understanding biological systems at system level [1]. For this relatively new field of biology, the behavior of a system cannot be explained by its components alone but it is necessary to examine the cellular dynamics and the mechanism processes. Thus, in order to understand biological systems, the interactions between components need to be studied and the way in which they give rise to the behavior of the whole system has to be analyzed. The aim of this paper is to propose a new system biology approach, bymeans of the kinetic theory for active particles [2]. In a recent paper [3], a critical analysis of the development of this theory has been presented, with the aim to capture the peculiar characteristics of the evolution of living systems, possibly toward a mathematical theory of evolution. This goal can be achieved by transferring the phenomenological analysis offered by anthropologists [4–6] into the formal description offered by equations derived within the framework of mathematical sciences. The critical discussion that followed [3] has put in evidence [7–12] that the interplay between mathematics and biology should be focused on the complexity features of living systems. In other words, mathematics should be able to retain, as far as possible, this crucial aspect. The latter appears as an obliged passage to pursue the objective of what is considered one of the greatest scientific revolutions that will hopefully characterize this century, namely the mathematical formalization of biology [13]. This effort definitely needs a great deal of research activity and human energy considering that it has to overcome the conceptual difficulties of the lack of first principles, as critically analyzed in various papers ([14–16] among others). Although the present state-of-the-art is still far from the aforementioned ambitious objective, any contribution, to even small progresses in that direction, is a challenging opportunity for appliedmathematicians. This is the objective of this paper, that is devoted to design a modeling approach and to identify the mathematical tools that can be achieved to deal with it. ✩ Partially supported by the European Union FP7 Health Research Grant number FP7-HEALTH-F4-2008-202047. ∗ Corresponding author. Tel.: +39 011 564 7514; fax: +39 011 564 7599. E-mail addresses: [email protected] (V. Coscia), [email protected] (L. Fermo), [email protected] (N. Bellomo). 0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.09.043 V. Coscia et al. / Computers and Mathematics with Applications 62 (2011) 3902–3911 3903 The sequential steps of the approach, as we shall see, are the following. (i) Assessment of the complexity features of biological systems in general, followed by the development of amathematical structure suitable to retain the aforementioned features. (ii) Identification of the scales that are appropriate to represent the specific system under consideration. (iii) Development of a system biology approach, whose first step is the decomposition of the overall system into functional subsystems. (iv) Derivation of the mathematical tools suitable to model, at each scale, the dynamics of the functional subsystems. (v) Modeling the interactions among the various components of the overall system, taking into account the networks and the multiscale aspects of their connection. The above project follows [17], where general topics on complexity andmathematical toolswere presented. The contents are organized as follows. Section 2 is devoted to the identification of the complexity features of living, and hence complex, systems. Section 3 reports the mathematical structure given by the kinetic theory for active particles [2] able to model such kind of systems and examines its consistency with the aforementioned key features. Section 4 focuses on themain objective of this paper, that is to propose the guidelines to develop amodeling approach for biological systems. Hence, at first a general methodology to reduce the complexity is presented, and then appropriate mathematical tools are introduced. Section 5 critically analyzes discrete setting of the variables at the microscopic scale. The aim is twofold, namely, as a computational tool and as an approach to model genetic mutations. Section 6 looks ahead to research perspectives starting from a deep insight to multiscale issues and following by some speculations related to conceivable paths to develop a mathematical theory for biological systems. 2. On the complexity of biological systems Biological systems are very different from the physical ones. In fact, while the latter are composed bymany copies of few elements, the former ones are constituted by a large variety of components: biological systems contain frommillions to a few copies of each of thousands of different elements, and this is one of the most important characteristic of such systems. They are constituted by living entities which have the ability to develop a specific strategy and an organizing ability, depending on the state of the surrounding environment. This strategy can be expressed without the application of any external organizing principle and depends on the search of individuals for their best fitness, sometimes just for their survival. In various cases such a skill evolves in time. In fact, living systems receive inputs from the environment and have the ability to learn from past experience, in order to adapt themselves to the changing-in-time external conditions [6]. This strategy is not the same for all entities. Indeed, heterogeneity characterizes a great part of living systems. Interacting entities can appear and behave different to many extent, even though they share the same molecular structure, for instance due to different phenotype expression generated by the same genotype. Moreover, living entities typically operate out-of-equilibrium. For example, a constant struggle against the environment is developed to remain in a particular out-of-equilibrium state, namely stay alive [18]. Interactions contribute to the development of the aforementioned strategy. These are nonlinear and involve immediate neighbors, but in some cases also distant entities. This is what happens at the level of cells, which have the ability to communicate by signaling and can choose different observation paths within networks that evolve in time. Living entities play a game at each interaction with an output that is technically related to their strategy often associated to the surviving and the adaptation ability. The presence of this strategy produces mutations and selections given by destructive and/or proliferative events. Moreover, all living systems are evolutionary: birth processes can generate individuals that fit better the outer environment, which in turn generate new ones fitting better and better. In conclusion, such kind of systems present a great complexity and if wewant tomodel themwemust handle this aspect. From amathematical point of view, the complexity is translated in a large number of variables, and hence in a large number of equations able to describe the overall system. On the other hand, this implies an high computational effort. Consequently, to model such kind of systems at first we have to reduce this complexity. An additional difficulty arises when we observe that the study of biological systems needs a multiscale approach. For instance, the dynamics at the molecular (genetic) level determines the cellular behaviors; moreover, the structure of macroscopic tissues depends on such dynamics. In the following section, we will see that a modeling approach for these kind of systems is actually possible. As already mentioned, the first step will be to reduce complexity, while the second one will be to specialize a mathematical structure able to model such systems. 3. On the kinetic theory for active particles This section deals with the mathematical approach we follow to describe the evolution in time of the biological systems under consideration. More precisely, the mathematical method we will adopt is that suggested by the kinetic theory for active particles, briefly the KTAP theory. 3904 V. Coscia et al. / Computers and Mathematics with Applications 62 (2011) 3902–3911 Introduced in [2] and further developed in [19,20], such a theory allows to model living systems characterized by the following five features. (i) The system is made up of a large number of interacting entities, called active particles, whose physical microscopic state is described by a set of variables. Among the others, a variable called activity represents the individual ability to express a specific strategy. (ii) The activity variable is heterogeneously distributed over the active particles. Thismeans that the entities can differ even if they have the same structure. Interactions modify the state of the particles, while the strategy they express can be modified by the shape of their heterogeneous distribution. (iii) Interactions involve not only immediate neighbors (short range interactions) but also the distant ones (long range interactions). Indeed, living systems communicate each other directly or through media. Consequently, each entity interacts with all the others in a domain whose elements are able to communicate. In some cases, such a domain is identified with the visibility zone, in other cases with a communication network. (iv) Interactions are complex, namely the overall output of the game that an active particle plays with the ones lying in its interaction domain is not the linear superposition of its separated interactions with all of them, but a complex combination whose form depends on the strategy that all particles can develop. (v) The output of the gamemodifies the activity of interacting particles andmay also generate, in the proliferative process, particles with a different structure (for instance, entities with a different phenotype). The KTAP theory has been applied by various authors to model complex systems in life sciences, for instance in immune competition [21–25], social dynamics [26], spread of epidemics [27,28], interpretation of clinical data [29], andmany others. These modeling approaches are based on linearly additive interactions but recently the modeling of nonlinear interactions has been investigated [30] and applied to vehicular traffic [31] and crowd dynamics [32]. In order to model living systems, the KTAP theory requires at first that all particles expressing the same strategy are organized in the same functional subsystem. Consequently, the system under consideration is divided into subsystems that, from now on, we assume to be in number of n. For each of them, a probability distribution function is introduced fi = fi(t, u) : [0, T ] × Du → R, i = 1, . . . , n (3.1) where the index i denotes the subsystem and u ∈ Du is the activity variable. The quantity fi(t, u)du represents the number of particles whose state, at time t , is in the elementary volume [u, u + du] and consequently,