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Mathematics toward systems biology and complexity Reply to comments on “On the interplay between mathematics and biology – Hallmarks toward a new systems biology” Nicola Bellomo ∗ , Ahmed Elaiw, Abdullah M. Althiabi, Mohammed Ali Alghamdi Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia Received 12 February 2015; accepted 12 February 2015 Available online 3 March 2015 Communicated by J. Fontanari
1. Introduction This paper provides a reply to the comments on paper [1] proposed by various authors. The contents are not limited to a technical issues. In fact, further speculations and reasonings have been stimulated by those comments, which deserve attention also in view of future research programs to cover aspects of the general theory, which has not yet fully developed. Indeed, we do not naively claim that the theory reviewed in [1] is exhaustive, as it needs to be regarded as a first step toward the development of a new systems biology. The main contribution of [1] is a review and critical analysis of kinetic models of cellular dynamics from the very first research papers, see the review delivered by the book [2], – where interactions are modeled by binary games, to recent developments, – where nonlinear multiple interactions with mutations and selections are included [3,4]. The conceptual framework, from the side of mathematical sciences is given in [3], which also shows how the theoretical approach can be applied to the derivation of a variety of models. Subsequently, a framework, where interactions with an evolving external environment are accounted for, have been derived in paper [4]. A unified revisiting is given in [1]. The various comments examined in the following partially overlap on the same topics, therefore we decided to present our reply focusing on topics rather than on authors, whose comments are, however, referred to the afore mentioned topics. Bearing all above in mind, the following issues are discussed in the next sections: • Modeling interactions at the scale of cells and relation of the mathematical structures with those of the classical kinetic theory; • Development of a system theory approach; DOI of original article: http://dx.doi.org/10.1016/j.plrev.2014.12.002. DOIs of comments: http://dx.doi.org/10.1016/j.plrev.2015.01.012, http://dx.doi.org/10.1016/j.plrev.2015.01.004, http://dx.doi.org/10.1016/j.plrev.2015.02.003, http://dx.doi.org/10.1016/j.plrev.2015.01.001, http://dx.doi.org/10.1016/j.plrev.2015.02.004, http://dx.doi.org/10.1016/j.plrev.2015.01.005, http://dx.doi.org/10.1016/j.plrev.2015.01.024, http://dx.doi.org/10.1016/j.plrev.2015.01.015, http://dx.doi.org/10.1016/j.plrev.2015.01.009. * Corresponding author. E-mail addresses: [email protected] (N. Bellomo), [email protected] (A. Elaiw), [email protected] (A.M. Althiabi), [email protected] (M.A. Alghamdi). http://dx.doi.org/10.1016/j.plrev.2015.02.006 1571-0645/© 2015 Published by Elsevier B.V.
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• Foundations of biology focused on the specific phenomena considered in [1]; • Challenging analytic and computational problems. The authors of [1] are grateful to the authors comments [6–14] as their contents can be viewed as a further important hint to future speculations and research ideas. 2. Interactions and classical kinetic theory Interactions in the mathematical structure reviewed and critically analyzed in [1] are modeled, as observed in [11], by the right-hand-side term of the integro-differential system (7)–(19). The space structure is related to a network of interconnected nodes, where, in each of them, the dynamics is simplified to the case of space homogeneity. On the other hand interactions, as observed in [11], are nonlocal and nonlinearly additive. Moreover, interactions are not binary, but each particle, e.g. a cell, interacts with all particles, e.g. cells, in its quorum sensitivity domain. The output of interactions are modeled by theoretical tools of evolutive game theory. Therefore, the term “stochastic games” is used to account for the random state of the interacting entities, as well as for the probabilistic rules of the interaction. Migrations from one node to the other, which is an important feature of biological systems, can be taken into account for instance following the hallmarks proposed in [15,16]. A detailed analysis of interactions and the related implications on the derivation of mathematical structures is treated in [5], where the aforementioned features of interactions are presented. Of course, when dealing with the spatially homogeneous case, this specific feature might hide multiple interactions. In principles, space non-homogeneity in each node can be taken into account, however one has to tackle the conceptual difficulty of modeling interactions in a network of the class of equations proposed in [1]. It is worth stressing that one of the drawbacks of several models of the kinetic type approach, when applied to modeling living systems, is the assumption of binary interactions, as it is observed in [9]. However, this is not the case of the so-called KTAP theory [5], which includes nonlinearity, multiple interactions with mutations and selection. Of course, this specific feature appears clearly for space-dependent models, such as models of traffic and swarming [17,18], but it is hidden in the case of space homogeneity. This topic is well understood in the comment by Gibelli [10], who starts from the analysis of interactions to understand the strategy to obtain quantitative results. Therefore, we feel comfortable to state that the crucial problem of modeling interactions has now a satisfactory answer [5] and has already been applied in a variety of case studies, for example vehicular traffic [18], crowds [19], and swarms [17]. Once the main features of interactions have been properly put in evidence, the problem of connection with classical kinetic models, posed in [11], but also in [6] related to the time and space scaling, deserves attention and further analysis. Therefore, let us look again at the mathematical structures reviewed in [1] out of the contents of [3] and [4]. Our remark is that classical kinetic theory does not include the mathematical structures under consideration; in fact interactions are nonlinearly additive and are not reversible, namely living systems share very little with classical particles. On the other hand, one might argue on alternative reasonings, namely do the structures of [1] include, as a special case, classical kinetic models? A first answer to this query is given in [20], where the class of equations is far less general than the one in [1]. Therefore, this topic has to be regarded as a hint for future research activity. 3. On a systems theory approach The idea of developing a systems biology approach based on evolution, mutations and selection is a key hallmark not only of [1], but also of the development of future interdisciplinary research activity, which will capture a great deal of energy of scientists not only in the field of biology, but also in mathematics, physics, informatics, chemistry, etc., in an interdisciplinary effort, where the scientific contributions in different fields should be addressed toward this challenging field. Therefore, we have appreciated the contributions of [6] and [14], who have well understood that such a problem is open to future speculations and contribution, while the topics treated in [1] offer a mathematical structure to be properly specialized in specific cases. The need of new structures is nowadays well accepted by the scientific community in general [21], but also by mathematicians [22]. Indeed, the contribution of [1] is mainly focused on this topic looking ahead to future perspectives.
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The most important one, according to the authors’ bias, consists in setting specific modes, derived within the framework of the said structure, in networks [23,24] corresponding to biological systems [25,26]. Recent contributions on the networking of models of biology, such as [27], can be further developed for more general mathematical structures. Bearing all above in mind, it is worth stressing that an important dynamics, whose study needs coupling specific models and networks, is that of migration phenomena see, for example [28–30]. Networking of complex models appears to be an important field of research. In some cases, such as vehicular traffic, it has been treated in a satisfactory way [31], while the more complex features of biological systems need additional studies. It is worth stressing that the structure under consideration has the ability of capturing the dynamics of mutations and selection, which appear not only at cellular level in all genetic diseases [32–35], but also in the case of virus mutations [36]. No wonder that this topic attracted by applied mathematicians [4,37] interested to formalize the complex rules of evolution. But, how this type of models can be inserted in complex networks? At present a satisfactory answer cannot yet be given, simply this topic can be indicated as a challenging research perspective. 4. Foundations of biology Some comments, e.g. [6,14], account for the need of addressing the structure toward specific biological phenomena. Comments [12] and [13] focus on problems of nonlinear diffusion and pattern formation. The literature on this topic witnesses a long story of research activity in the field motivated also by philosophical addresses such as the essay [38]. An important reference in biology is the celebrated model by Keller and Segel, which attracted the attention of several mathematicians, see the essays and surveys [39–41], who proved a variety of interesting problems reviewed in [39]. Let us now focus more precisely on the problem raised in [12] and discuss how far the structures presented in [1] are consistent with nonlinear diffusion and pattern formation. The answer, limited however, to closed systems, is given in a sequel of papers [42–44], where it is shown how macroscopic models of the aforementioned phenomena can be derived from particular specializations of the said structure [1]. Indeed, we do agree that nonlinear diffusion and pattern formation should march together, however, the study of this type of problems for open systems and in the presence of post-Darwinian mutations and selection is a field still waiting for future research contributions. All biological systems, even the simplest ones, have to be treated by multiscale methods from genes to organs, through cells tissues, and finally their organization into organs, which collectively express biological functions. Therefore, it is impossible examining a biological phenomenon focusing on one scale only. Mathematics can offer conceptual methods, which have to be implemented to the study of real problems. This is a task as much difficult as the derivation of theoretical methods. 5. Challenging analytic and computational problems Comment [7] is mainly focused on two topics, which deserve attention: Analytic problems generated by models derived within the frameworks of [1] and the possible of using that mathematical structure in fields somehow different from biology. While we agree with this comment, we observe that one of the most challenging analytic problem is the derivation of macroscopic models from the underlying description at the microscopic scale. This topic has already been discussed in the preceding section referring to [42–44], where also some research perspectives have been indicated. Similar reasonings can be developed for the qualitative analysis of mathematical problems, where the most challenging topic is the proof of asymptotic behaviors, see [3,45]. Concerning the second topic a precise question is posed in [7], namely Can the authors address research activity in the field of social sciences or, more generally, in life sciences? More in detail, the author refers to [46–48]. This topic is the main one in the comment [8], who specifically refers to a well defined bibliography as well. We wish simply to state that the analogy is significant and that we do share the hint posed in [7], which is also posed by physicists and researchers in the field of social sciences [49,50]. Moreover one can look at recent developments from [51] to [52], as well as at approaches more directly related to kinetic theory [48]. Moreover, comment [8] focuses on a well defined issue, namely whether post-Darwinian theories can be applied to specific social systems. We can only agree with this idea, which is now also object of recent theories in economy and sociology often, and correctly, viewed as strongly interconnected fields. Moreover, we do think that all living systems
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have common features, hence mathematical sciences can extract ideas from models in a certain field and address them, after appropriate developments, to a different one. The contribution of mathematics cannot be limited to qualitative analysis of solutions [53], as computational problems involve non-trivial difficulties [54]. Some reasonings on this topic are proposed in [10], where it is remarked that classical deterministic methods are not appropriate to obtain simulations for the class of equations proposed in [1]. We do agree with this remark as it is true that suitable developments of particles methods [48] are more appropriate to capture the dynamics of interactions. Finally, let us state the most challenging problem appears to be developing the full scale approach from genes to tissue. This means understanding the dynamics at the molecular scale and how the dynamics of gene expression is transferred to that of cells and hence to tissues. Although specific contributions are known in the literature, e.g. [55–57], the formalization of the whole process still needs to be developed. Understanding this complex dynamics might lead, in the authors’ opinion, to a deeper understanding of the complexity of the immune competition [58,59]. This topic is presented in [1], but it is not yet fully understood. However, a conjecture is there proposed, namely that the mathematical structures proposed in [1] can be used also to model the interactive dynamics between the molecular scale of genes and that of cells. We do believe that this conjecture deserves attention and needs to be further investigated by taking advantage of a literature gradually moving toward such challenging objective, which, if correct, can explain several biological phenomena at present not fully understood. Acknowledgement This research is supported Deanship by Scientific Research (DSR), King Abdulaziz University of Saudi Arabia, Jeddah, under grant No. (98-130-36-HiCi). The Authors, therefore, acknowledge with thanks DSR technical and financial support. References [1] Bellomo N, Elaiw A, Althiabi AM, Alghamdi A. On the interplay between mathematics and biology. Hallmarks toward a new systems biology. Phys Life Rev 2015;12:44–64 [in this issue]. [2] Bellouquid A, Delitala M. Mathematical methods and tools of kinetic theory towards modelling complex biological systems. Math Models Methods Appl Sci 2005;15:1639–66. [3] Bellouquid A, De Angelis E, Knopoff D. From the modeling of the immune hallmarks of cancer to a black swan in biology. Math Models Methods Appl Sci 2013;23:949–78. [4] De Angelis E. On the mathematical theory of post-darwinian mutations, selection, and evolution. Math Models Methods Appl Sci 2014;24:2723–42. [5] Bellomo N, Knopoff D, Soler J. On the difficult interplay between life, complexity, and mathematical sciences. Math Models Methods Appl Sci 2013;23:1861–913. [6] Banasiak J. Multi-scale problems in complex domains – a mathematical framework for systems biology: Comment on “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo, et al. Phys Life Rev 2015;12:65–7 [in this issue]. [7] Bellouquid A. From systems biology to analytic problems: Comment on the paper “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo, et al. Phys Life Rev 2015;12:68–9 [in this issue]. [8] Dolfin M. How far systems biology methods can contribute to understand social systems?: Comment on the paper “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo, et al. Phys Life Rev 2015;12:70–1 [in this issue]. [9] Eftimie R. The quest for a new modeling framework in mathematical biology: Comment on the paper “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo, et al. Phys Life Rev 2015;12:72–3 [in this issue]. [10] Gibelli L. Stochastic features and strategy of computational methods: Comment on the paper “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo, et al. Phys Life Rev 2015;12:74–5 [in this issue]. [11] Lachowicz M, Szyma´nska Z. Nonlocal models of biological phenomena: Comment on “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo, et al. Phys Life Rev 2015;12:76–7 [in this issue]. [12] López JL, Soler J. Mathematics and biology: a round trip: Comment on “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo et al. Phys Life Rev 2015;12:78–80 [in this issue]. [13] Nieto J. The kinetic theory of active particles as a biological systems approach: Comment on “On the interplay between mathematics and biology. Hallmarks toward a new systems biology” by N. Bellomo et al. Phys Life Rev 2015;12:81–2 [in this issue]. [14] Surulescu C. New trends in mathematical biology: from the subcellular scale to cell populations and tissues: Comment on “On the interplay between mathematics and biology. Hallmarks toward a new systems biology. Phys Life Rev 2015;12:83–4 [in this issue]. [15] Knopoff D. On the modeling of migration phenomena on small networks. Math Models Methods Appl Sci 2013;23:541–63. [16] Knopoff D. On a mathematical theory of complex systems on networks with application to opinion formation. Math Models Methods Appl Sci 2014;24:405–26.
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[17] Bellomo N, Soler J. On the mathematical theory of the dynamics of swarms viewed as complex systems. Math Models Methods Appl Sci 2012;22:1140006. [18] Bellomo N, Bellouquid A, Nieto J, Soler J. On the multiscale modeling of vehicular traffic: from kinetic to hydrodynamics. Discrete Contin Dyn Syst, Ser B 2014;19:1869–88. [19] Bellomo N, Piccoli B, Tosin A. Modeling crowd dynamics from a complex system viewpoint. Math Models Methods Appl Sci 2012:22. Paper No. 1230004. [20] Arlotti L, Bellomo N, De Angelis E. Generalized kinetic (Boltzmann) models: mathematical structures and applications. Math Models Methods Appl Sci 2002;12:567–91. [21] Hartwell HL, Hopfield JJ, Leibner S, Murray AW. From molecular to modular cell biology. Nature Rev 1999;402:c47–52. [22] Gromov M. In a search for a structure, Part 1: On entropy, http://www.ihes.fr/~gromov/PDF/structre-serch-entropy-july5-2-2012.pdf, July, 2012 [Online, accessed 16-July-2014]. [23] Barabási L. The science of networks. Cambridge: Perseus; 2002. [24] Barrat A, Bathélemy M, Vespignani A. The structure and dynamics of networks. Princeton: Princeton University Press; 2006. [25] Komarova N, Sengupta A, Nowak MA. Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal, instability. J Theor Biol 2003;223:433–50. [26] Komarova N, Wodarz D. Targeted cancer treatment in silico. Modeling and simulation in science, engineering and technology. Boston: Birkhäuser; 2014. [27] Borsche R, Göttlich SA, Klar A, Schillen P. The scalar Keller and Segel on networks. Math Models Methods Appl Sci 2014;24:221–47. [28] Fontanari JF, Serva M. Effect of migration in a diffusion model for template coexistence in protocells. Bull Math Biol 2014;76:654–72. [29] Lorenz T, Surulescu C. On a class of multiscale cancer cell migration models: well-posedness in less regular function spaces. Math Models Methods Appl Sci 2014;24:2383–436. [30] Kelkel J, Surulescu C. A multiscale approach to cell migration in tissue networks. Math Models Methods Appl Sci 2012;22:1150017. [31] Fermo L, Tosin A. A fully-discrete-state kinetic theory approach to traffic flow on road networks. Math Models Methods Appl Sci 2015;25:423–61. [32] Fontanari JF, Serva M. Nonlinear group survival in Kimura’s model for the evolution of altruism. Math Biosci 2014;249:18–26. [33] Nesse RM, Bergstrom CT, Ellison PT, Flier JS, Gluckman P, Govindaraju DR, et al. Evolution in health and medicine Sackler colloquium: making evolutionary biology a basic science for medicine. Proc Natl Acad Sci USA 2010;107:1800–7. [34] Perlman RL. Evolutionary biology: a basic science for medicine in the 21st century. Perspect Biol Med 2011;54:75–88. [35] Woese CR. A new biology for a new century. Microbiol Mol Biol Rev 2004;68:173–86. [36] De Lillo S, Delitala M, Salvatori MC. Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles. Math Models Methods Appl Sci 2009;19:1405–26. [37] Lorz A, Mirrahimi S, Perthame B. Dirac mass dynamics in multidimensional nonlocal parabolic equations. Commun Partial Differ Equ 2011;36:1071–98. [38] Roth S. Mathematics and biology: a Kantian view of the history of pattern formation biology. Dev Genes Evol 2011;221:255–79. [39] Bellomo N, Bellouquid A, Tao Y, Winkler M. Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math Models Methods Appl Sci 2015:25. [40] Hillen T, Painter KJ. A users guide to PDE models for chemotaxis. J Math Biol 2009;58:183–217. [41] Horstmann D. From 1970 until present: the Keller–Segel model in chemotaxis and its consequences: Jahresber I. Jahresber Dtsch Math-Ver 2003;105:103–65. [42] Bellomo N, Bellouquid A, Nieto J, Soler J. Multicellular biological growing systems: hyperbolic limits towards macroscopic description. Math Models Methods Appl Sci 2007;17:1675–92. [43] Bellomo N, Bellouquid A, Nieto J, Soler J. Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems. Math Models Methods Appl Sci 2010;20:1179–207. [44] Bellomo N, Bellouquid A, Nieto J, Soler J. On the asymptotic theory from microscopic to macroscopic tissue models: an overview with perspectives. Math Models Methods Appl Sci 2012:22. Paper No. 1130001. [45] Bellouquid A, CH-Chaoui M. Asymptotic analysis of a nonlinear integro-differential system modeling the immune response. Comput Math Appl 2014;68:905–14. [46] Baley KD. Sociology and new systems theory – toward a theoretical synthesis. Suny Press; 1994. [47] Bellomo N, Herrero MA, Tosin A. On the dynamics of social conflicts: looking for the black swan. Kinet Relat Models 2013;6:459–79. [48] Pareschi L, Toscani G. Interacting multiagent systems: kinetic equations and Monte Carlo methods. Oxford University Press; 2013. [49] Galam S. Sociophysics. Springer; 2013. [50] Bonacich P, Lu P. Introduction to mathematical sociology. Princeton University Press; 2012. [51] Ajmone Marsan G, Bellomo N, Egidi M. Towards a mathematical theory of complex socio-economical systems by functional subsystems representation. Kinet Relat Models 2008;1:249–78. [52] Dolfin M, Lachowicz M. Modeling altruism and selfishness in welfare dynamics: the role of nonlinear interactions. Math Models Methods Appl Sci 2014;24:2361–81. [53] Arlotti L, De Angelis E, Fermo L, Lachowicz M, Bellomo N. On a class of integro-differential equations modeling complex systems with nonlinear interactions. Appl Math Lett 2012;25:490–5. [54] Ghiroldi GP, Gibelli L. A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization. J Comput Phys 2014;258:568–84. [55] Calvo J, Nieto J, Soler J, Vásquez MO. On a dispersive model for the unzipping of double-stranded DNA molecules. Math Models Methods Appl Sci 2014;24:495–511.
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ScienceDirect Physics of Life Reviews 12 (2015) 85–90 www.elsevier.com/locate/plrev
Reply to comment
Mathematics toward systems biology and complexity Reply to comments on “On the interplay between mathematics and biology – Hallmarks toward a new systems biology” Nicola Bellomo ∗ , Ahmed Elaiw, Abdullah M. Althiabi, Mohammed Ali Alghamdi Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia Received 12 February 2015; accepted 12 February 2015 Available online 3 March 2015 Communicated by J. Fontanari
1. Introduction This paper provides a reply to the comments on paper [1] proposed by various authors. The contents are not limited to a technical issues. In fact, further speculations and reasonings have been stimulated by those comments, which deserve attention also in view of future research programs to cover aspects of the general theory, which has not yet fully developed. Indeed, we do not naively claim that the theory reviewed in [1] is exhaustive, as it needs to be regarded as a first step toward the development of a new systems biology. The main contribution of [1] is a review and critical analysis of kinetic models of cellular dynamics from the very first research papers, see the review delivered by the book [2], – where interactions are modeled by binary games, to recent developments, – where nonlinear multiple interactions with mutations and selections are included [3,4]. The conceptual framework, from the side of mathematical sciences is given in [3], which also shows how the theoretical approach can be applied to the derivation of a variety of models. Subsequently, a framework, where interactions with an evolving external environment are accounted for, have been derived in paper [4]. A unified revisiting is given in [1]. The various comments examined in the following partially overlap on the same topics, therefore we decided to present our reply focusing on topics rather than on authors, whose comments are, however, referred to the afore mentioned topics. Bearing all above in mind, the following issues are discussed in the next sections: • Modeling interactions at the scale of cells and relation of the mathematical structures with those of the classical kinetic theory; • Development of a system theory approach; DOI of original article: http://dx.doi.org/10.1016/j.plrev.2014.12.002. DOIs of comments: http://dx.doi.org/10.1016/j.plrev.2015.01.012, http://dx.doi.org/10.1016/j.plrev.2015.01.004, http://dx.doi.org/10.1016/j.plrev.2015.02.003, http://dx.doi.org/10.1016/j.plrev.2015.01.001, http://dx.doi.org/10.1016/j.plrev.2015.02.004, http://dx.doi.org/10.1016/j.plrev.2015.01.005, http://dx.doi.org/10.1016/j.plrev.2015.01.024, http://dx.doi.org/10.1016/j.plrev.2015.01.015, http://dx.doi.org/10.1016/j.plrev.2015.01.009. * Corresponding author. E-mail addresses: [email protected] (N. Bellomo), [email protected] (A. Elaiw), [email protected] (A.M. Althiabi), [email protected] (M.A. Alghamdi). http://dx.doi.org/10.1016/j.plrev.2015.02.006 1571-0645/© 2015 Published by Elsevier B.V.
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• Foundations of biology focused on the specific phenomena considered in [1]; • Challenging analytic and computational problems. The authors of [1] are grateful to the authors comments [6–14] as their contents can be viewed as a further important hint to future speculations and research ideas. 2. Interactions and classical kinetic theory Interactions in the mathematical structure reviewed and critically analyzed in [1] are modeled, as observed in [11], by the right-hand-side term of the integro-differential system (7)–(19). The space structure is related to a network of interconnected nodes, where, in each of them, the dynamics is simplified to the case of space homogeneity. On the other hand interactions, as observed in [11], are nonlocal and nonlinearly additive. Moreover, interactions are not binary, but each particle, e.g. a cell, interacts with all particles, e.g. cells, in its quorum sensitivity domain. The output of interactions are modeled by theoretical tools of evolutive game theory. Therefore, the term “stochastic games” is used to account for the random state of the interacting entities, as well as for the probabilistic rules of the interaction. Migrations from one node to the other, which is an important feature of biological systems, can be taken into account for instance following the hallmarks proposed in [15,16]. A detailed analysis of interactions and the related implications on the derivation of mathematical structures is treated in [5], where the aforementioned features of interactions are presented. Of course, when dealing with the spatially homogeneous case, this specific feature might hide multiple interactions. In principles, space non-homogeneity in each node can be taken into account, however one has to tackle the conceptual difficulty of modeling interactions in a network of the class of equations proposed in [1]. It is worth stressing that one of the drawbacks of several models of the kinetic type approach, when applied to modeling living systems, is the assumption of binary interactions, as it is observed in [9]. However, this is not the case of the so-called KTAP theory [5], which includes nonlinearity, multiple interactions with mutations and selection. Of course, this specific feature appears clearly for space-dependent models, such as models of traffic and swarming [17,18], but it is hidden in the case of space homogeneity. This topic is well understood in the comment by Gibelli [10], who starts from the analysis of interactions to understand the strategy to obtain quantitative results. Therefore, we feel comfortable to state that the crucial problem of modeling interactions has now a satisfactory answer [5] and has already been applied in a variety of case studies, for example vehicular traffic [18], crowds [19], and swarms [17]. Once the main features of interactions have been properly put in evidence, the problem of connection with classical kinetic models, posed in [11], but also in [6] related to the time and space scaling, deserves attention and further analysis. Therefore, let us look again at the mathematical structures reviewed in [1] out of the contents of [3] and [4]. Our remark is that classical kinetic theory does not include the mathematical structures under consideration; in fact interactions are nonlinearly additive and are not reversible, namely living systems share very little with classical particles. On the other hand, one might argue on alternative reasonings, namely do the structures of [1] include, as a special case, classical kinetic models? A first answer to this query is given in [20], where the class of equations is far less general than the one in [1]. Therefore, this topic has to be regarded as a hint for future research activity. 3. On a systems theory approach The idea of developing a systems biology approach based on evolution, mutations and selection is a key hallmark not only of [1], but also of the development of future interdisciplinary research activity, which will capture a great deal of energy of scientists not only in the field of biology, but also in mathematics, physics, informatics, chemistry, etc., in an interdisciplinary effort, where the scientific contributions in different fields should be addressed toward this challenging field. Therefore, we have appreciated the contributions of [6] and [14], who have well understood that such a problem is open to future speculations and contribution, while the topics treated in [1] offer a mathematical structure to be properly specialized in specific cases. The need of new structures is nowadays well accepted by the scientific community in general [21], but also by mathematicians [22]. Indeed, the contribution of [1] is mainly focused on this topic looking ahead to future perspectives.
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The most important one, according to the authors’ bias, consists in setting specific modes, derived within the framework of the said structure, in networks [23,24] corresponding to biological systems [25,26]. Recent contributions on the networking of models of biology, such as [27], can be further developed for more general mathematical structures. Bearing all above in mind, it is worth stressing that an important dynamics, whose study needs coupling specific models and networks, is that of migration phenomena see, for example [28–30]. Networking of complex models appears to be an important field of research. In some cases, such as vehicular traffic, it has been treated in a satisfactory way [31], while the more complex features of biological systems need additional studies. It is worth stressing that the structure under consideration has the ability of capturing the dynamics of mutations and selection, which appear not only at cellular level in all genetic diseases [32–35], but also in the case of virus mutations [36]. No wonder that this topic attracted by applied mathematicians [4,37] interested to formalize the complex rules of evolution. But, how this type of models can be inserted in complex networks? At present a satisfactory answer cannot yet be given, simply this topic can be indicated as a challenging research perspective. 4. Foundations of biology Some comments, e.g. [6,14], account for the need of addressing the structure toward specific biological phenomena. Comments [12] and [13] focus on problems of nonlinear diffusion and pattern formation. The literature on this topic witnesses a long story of research activity in the field motivated also by philosophical addresses such as the essay [38]. An important reference in biology is the celebrated model by Keller and Segel, which attracted the attention of several mathematicians, see the essays and surveys [39–41], who proved a variety of interesting problems reviewed in [39]. Let us now focus more precisely on the problem raised in [12] and discuss how far the structures presented in [1] are consistent with nonlinear diffusion and pattern formation. The answer, limited however, to closed systems, is given in a sequel of papers [42–44], where it is shown how macroscopic models of the aforementioned phenomena can be derived from particular specializations of the said structure [1]. Indeed, we do agree that nonlinear diffusion and pattern formation should march together, however, the study of this type of problems for open systems and in the presence of post-Darwinian mutations and selection is a field still waiting for future research contributions. All biological systems, even the simplest ones, have to be treated by multiscale methods from genes to organs, through cells tissues, and finally their organization into organs, which collectively express biological functions. Therefore, it is impossible examining a biological phenomenon focusing on one scale only. Mathematics can offer conceptual methods, which have to be implemented to the study of real problems. This is a task as much difficult as the derivation of theoretical methods. 5. Challenging analytic and computational problems Comment [7] is mainly focused on two topics, which deserve attention: Analytic problems generated by models derived within the frameworks of [1] and the possible of using that mathematical structure in fields somehow different from biology. While we agree with this comment, we observe that one of the most challenging analytic problem is the derivation of macroscopic models from the underlying description at the microscopic scale. This topic has already been discussed in the preceding section referring to [42–44], where also some research perspectives have been indicated. Similar reasonings can be developed for the qualitative analysis of mathematical problems, where the most challenging topic is the proof of asymptotic behaviors, see [3,45]. Concerning the second topic a precise question is posed in [7], namely Can the authors address research activity in the field of social sciences or, more generally, in life sciences? More in detail, the author refers to [46–48]. This topic is the main one in the comment [8], who specifically refers to a well defined bibliography as well. We wish simply to state that the analogy is significant and that we do share the hint posed in [7], which is also posed by physicists and researchers in the field of social sciences [49,50]. Moreover one can look at recent developments from [51] to [52], as well as at approaches more directly related to kinetic theory [48]. Moreover, comment [8] focuses on a well defined issue, namely whether post-Darwinian theories can be applied to specific social systems. We can only agree with this idea, which is now also object of recent theories in economy and sociology often, and correctly, viewed as strongly interconnected fields. Moreover, we do think that all living systems
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have common features, hence mathematical sciences can extract ideas from models in a certain field and address them, after appropriate developments, to a different one. The contribution of mathematics cannot be limited to qualitative analysis of solutions [53], as computational problems involve non-trivial difficulties [54]. Some reasonings on this topic are proposed in [10], where it is remarked that classical deterministic methods are not appropriate to obtain simulations for the class of equations proposed in [1]. We do agree with this remark as it is true that suitable developments of particles methods [48] are more appropriate to capture the dynamics of interactions. Finally, let us state the most challenging problem appears to be developing the full scale approach from genes to tissue. This means understanding the dynamics at the molecular scale and how the dynamics of gene expression is transferred to that of cells and hence to tissues. Although specific contributions are known in the literature, e.g. [55–57], the formalization of the whole process still needs to be developed. Understanding this complex dynamics might lead, in the authors’ opinion, to a deeper understanding of the complexity of the immune competition [58,59]. This topic is presented in [1], but it is not yet fully understood. However, a conjecture is there proposed, namely that the mathematical structures proposed in [1] can be used also to model the interactive dynamics between the molecular scale of genes and that of cells. We do believe that this conjecture deserves attention and needs to be further investigated by taking advantage of a literature gradually moving toward such challenging objective, which, if correct, can explain several biological phenomena at present not fully understood. Acknowledgement This research is supported Deanship by Scientific Research (DSR), King Abdulaziz University of Saudi Arabia, Jeddah, under grant No. (98-130-36-HiCi). The Authors, therefore, acknowledge with thanks DSR technical and financial support. References [1] Bellomo N, Elaiw A, Althiabi AM, Alghamdi A. On the interplay between mathematics and biology. Hallmarks toward a new systems biology. Phys Life Rev 2015;12:44–64 [in this issue]. [2] Bellouquid A, Delitala M. Mathematical methods and tools of kinetic theory towards modelling complex biological systems. Math Models Methods Appl Sci 2005;15:1639–66. [3] Bellouquid A, De Angelis E, Knopoff D. From the modeling of the immune hallmarks of cancer to a black swan in biology. Math Models Methods Appl Sci 2013;23:949–78. [4] De Angelis E. On the mathematical theory of post-darwinian mutations, selection, and evolution. Math Models Methods Appl Sci 2014;24:2723–42. [5] Bellomo N, Knopoff D, Soler J. 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