Multiscale mathematical models for biological systems

Abstract

Life activities are extremely complex phenomena in nature. From molecular signaling regulation to multi-cellular tissue formation and so on, the biological system consists of multiple temporal, spatial and functional scales. Multiscale mathematical models have extensive applications in life science research due to their capacity of appropriately simulating the complex multiscale biological systems. Many mathematical methods, including deterministic methods, stochastic methods as well as discrete or rule-based methods, have been widely used for modeling biological systems. However, the models at single scale are not sufficient to simulate complex biological systems. Therefore, in this paper we give a survey of two multiscale modeling approaches for biological systems. One approach is continuous stochastic method that couples ordinary differential equations and stochastic differential equations; Another approach is hybrid continuous-discrete method that couples agent-based model with partial differential equations. We then introduce the applications of these multiscale modeling approaches in systems biology and look ahead to the future research.

Key words： Multi-scale model; partial differential equations; agent-based model; systems biology

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 62273364, 11871070 & 11371060) and the Guangdong Basic and Applied Basic Research Foundation (2020B1515020047).

References

1 Aldridge B B, Burke J M, Lauffenburger D A. . Physicochemical modelling of cell signalling pathways. Nat Cell Biol 2006; 8(11): 1195–1203

2 AlonU. An Introduction to Systems Biology: Design Principles of Biological Circuits. Boca Raton: Chapman & Hall/CRC, 2006

3 Anderson A R A. A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math Med Biol J IMA 2005; 22(2): 163–186

4 Anderson A R A, Chaplain M A J. Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 1998; 60(5): 857–899

5 Ballesta A, Lopez J, Popgeorgiev N. . Data-driven modeling of SRC control on the mitochondrial pathway of apoptosis: implication for anticancer therapy optimization. PLoS Comput Biol 2013; 9(4): e1003011

6 Bonabeau E. Agent-based modeling: Methods and techniques for simulating human systems. Proc Nat Acad Sci USA 2002; 99(Suppl 3): 7280–7287

7 ChapmanS J. Multiscale mathematical modelling in medicine and biology. In: Proceedings of the 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation: Interfacing Modeling and Simulation with Mathematcial and Computational Sciences, Anderssen R S, Bradock R D and Newham L T H eds. Cairns, Australia, 2009, 13–22

8 Checa S, Prendergast P J. A mechanobiological model for tissue differentiation that includes angiogenesis: a lattice-based modeling approach. Ann Biomed Eng 2009; 37(1): 129–145

9 Checa S, Prendergast P J. Effect of cell seeding and mechanical loading on vascularization and tissue formation inside a scaffold: a mechano-biological model using a lattice approach to simulate cell activity. J Biomech 2010; 43(5): 961–968

10 Cheng J, Zhang J, Wu Z. . Inferring microenvironmental regulation of gene expression from single-cell RNA sequencing data using scMLnet with an application to COVID-19. Briefings Bioinf 2021; 22(2): 988–1005

11 ChopardBDroz M. Cellular Automata Modeling of Physical Systems. Cambridge: Cambridge University Press, 1998

12 Chou T C. Derivation and properties of Michaelis-Menten type and Hill type equations for reference ligands. J Theor Biol 1976; 59(2): 253–276

13 Dada J O, Mendes P. Multi-scale modelling and simulation in systems biology. Inter Biol 2011; 3(2): 86–96

14 Ellis P M, Morzycki W, Melosky B. . The role of the epidermal growth factor receptor tyrosine kinase inhibitors as therapy for advanced, metastatic, and recurrent non-small-cell lung cancer: a Canadian national consensus statement. Curr Oncol 2009; 16(1): 27–48

15 Garfinkel A, Tintut Y, Petrasek D. . Pattern formation by vascular mesenchymal cells. Proc Nat Acad Sci USA 2004; 101(25): 9247–9250

16 Geris L, Reed A A C, Vander Sloten J. . Occurrence and treatment of bone atrophic non-unions investigated by an integrative approach. PLoS Comput Biol 2010; 6(9): e1000915

17 Geris L, Vander Sloten J, Van Oosterwyck H. In silico biology of bone modelling and remodelling: regeneration. Phil Trans Royal Soc A: Math Phys Eng Sci 2009; 367(1895): 2031–2053

18 Geris L, Vander Sloten J, Van Oosterwyck H. Connecting biology and mechanics in fracture healing: an integrated mathematical modeling framework for the study of nonunions. Biomech Model Mechanobiol 2010; 9(6): 713–724

19 Gierer A, Meinhardt H. A theory of biological pattern formation. Kybernetik 1972; 12(1): 30–39

20 Gillespie D T. Exact stochastic simulation of coupled chemical reactions. J Phys Chem 1977; 81(25): 2340–2361

21 Gillespie D T. Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 2007; 58(1): 35–55

22 Gordeladze J O, Reseland J E, Duroux-Richard I. . From stem cells to bone: phenotype acquisition, stabilization, and tissue engineering in animal models. ILAR J 2009; 51(1): 42–61

23 Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 2001; 43(3): 525–546

24 Kansal A R, Torquato S, Harsh IV G R. . Cellular automaton of idealized brain tumor growth dynamics. Biosystems 2000; 55(1/2/3): 119–127

25 Karplus M, Petsko G A. Molecular dynamics simulations in biology. Nature 1990; 347(6294): 631–639

26 Katare S, Bhan A, Caruthers J M. . A hybrid genetic algorithm for efficient parameter estimation of large kinetic models. Comp Chem Eng 2004; 28(12): 2569–2581

27 Kauffman S A. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 1969; 22(3): 437–467

28 KitanoH. Foundations of Systems Biology. Cambridge, MA: MIT Press, 2001

29 KlebanerF C. Introduction to Stochastic Calculus with Applications, 2nd ed. Singapore: Imperial College Press, 2005

30 Klepeis J L, Lindorff-Larsen K, Dror R O. . Long-timescale molecular dynamics simulations of protein structure and function. Curr Opin Struct Biol 2009; 19(2): 120–127

31 KlippELiebermeister WWierlingC, . Systems Biology: A Textbook, 1st ed. Weinheim: Wiley-VCH, 2009, 13–22

32 Komarova S V, Smith R J, Dixon S J. . Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling. Bone 2003; 33(2): 206–215

33 Komori T. Regulation of bone development and maintenance by Runx2. Front Biosci 2008; 13(3): 898–903

34 Lee J M, Gianchandani E P, Eddy J A. . Dynamic analysis of integrated signaling, metabolic, and regulatory networks. PLoS Comput Biol 2008; 4(5): e1000086

35 Legewie S, Blüthgen N, Herzel H. Mathematical modeling identifies inhibitors of apoptosis as mediators of positive feedback and bistability. PLoS Comput Biol 2006; 2(9): e120

36 Lemaire V, Tobin F L, Greller L D. . Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J Theor Biol 2004; 229(3): 293–309

37 Liang J, Qian H. Computational cellular dynamics based on the chemical master equation: a challenge for understanding complexity. J Comput Sci Tech 2010; 25(1): 154–168

38 Liang W, Zheng Y, Zhang J. . Multiscale modeling reveals angiogenesis-induced drug resistance in brain tumors and predicts a synergistic drug combination targeting EGFR and VEGFR pathways. BMC Bioinf 2019; 20(Suppl 7): 203

39 Lovett M, Lee K, Edwards A. . Vascularization strategies for tissue engineering. Tissue Eng Part B: Reviews 2009; 15(3): 353–370

40 Macklin P, McDougall S, Anderson A R. . Multiscale modelling and nonlinear simulation of vascular tumour growth. J Math Biol 2009; 58(4/5): 765–798

41 Michaelis L, Menten M L. Die kinetik der invertinwirkung. Biochem Z 1913; 49: 333–369

42 MortonK WMayers D F. Numerical Solution of Partial Differential Equations: An Introduction. Cambridge: Cambridge University Press, 2005

43 Neumann L, Pforr C, Beaudouin J. . Dynamics within the CD95 death-inducing signaling complex decide life and death of cells. Mol Syst Biol 2010; 6(1):

44 Nguyen L H, Annabi N, Nikkah M. . Vascularized bone tissue engineering: approaches for potential improvement. Tissue Eng Part B: Reviews 2009; 18(5): 363–382

45 NicolisGPrigogine I. Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations. New York: John Wiley & Sons, 1977

46 ØksendalB K. Stochastic Differential Equations: An Introduction with Applications, 5th ed. New York: Springer-Verlag, 2003, 61–72

47 Patel A A, Gawlinski E T, Lemieux S K. . A cellular automaton model of early tumor growth and invasion: the effects of native tissue vascularity and increased anaerobic tumor metabolism. J Theor Biol 2001; 213(3): 315–331

48 Pivonka P, Zimak J, Smith D W. . Model structure and control of bone remodeling: a theoretical study. Bone 2008; 43(2): 249–263

49 Pivonka P, Zimak J, Smith D W. . Theoretical investigation of the role of the RANK-RANKL-OPG system in bone remodeling. J Theor Biol 2010; 262(2): 306–316

50 Rejniak K A, Anderson A R A. Hybrid models of tumor growth. Wiley Interdiscip Rev: Syst Biol Med 2011; 3(1): 115–125

51 Sandino C, Checa S, Prendergast P J. . Simulation of angiogenesis and cell differentiation in a CaP scaffold subjected to compressive strains using a lattice modeling approach. Biomaterials 2009; 31(8): 2446–2452

52 Sanz-Herrera J A, Garcia-Aznar J, Doblare M. Micro-macro numerical modelling of bone regeneration in tissue engineering. Comput Meth Appl Mech Eng 2008; 197(33): 3092–3107

53 Sanz-Herrera J A, Garcia-Aznar J, Doblare M. Simulation of bone remodelling and bone ingrowth within scaffolds. Key Eng Mater 2008; 377: 225–273

54 Sanz-Herrera J A, García-Aznar J M, Doblaré M. A mathematical approach to bone tissue engineering. Phi Trans Roy Soc A: Math Phys Eng Sci 2009; 367(1895): 2055–2078

55 Schlatter R, Schmich K, Avalos Vizcarra I. . ON/OFF and beyond-a Boolean model of apoptosis. PLoS Comput Biol 2009; 5(12): e1000595

56 Stokes C L, Lauffenburger D A. Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis. J Theor Biol 1991; 152(3): 377–403

57 Sun W, Lal P. Recent development on computer aided tissue engineering—a review. Comput Meth Prog Biomed 2002; 67(2): 85–103

58 Sun X Q, Bao J G, Nelson K C. . Systems modeling of anti-apoptotic pathways in prostate cancer: psychological stress triggers a synergism pattern switch in drug combination therapy. PLoS Comput Biol 2013; 9(12): e1003358

59 Sun X Q, Bao J G, Shao Y Z. Mathematical modeling of therapy-induced cancer drug resistance: connecting cancer mechanisms to population survival rates. Sci Rep 2016; 6(1): 22498

60 Sun X Q, Hu B. Mathematical modeling and computational prediction of cancer drug resistance. Briefings Bioinf 2018; 19(6): 1382–1399

61 Sun X Q, Kang Y Q, Bao J G. . Modeling vascularized bone regeneration within a porous biodegradable CaP scaffold loaded with growth factors. Biomaterials 2013; 34(21): 4971–4981

62 Sun X Q, Su J, Bao J G. . Cytokine combination therapy prediction for bone remodeling in tissue engineering based on the intracellular signaling pathway. Biomaterials 2012; 33(33): 8265–8276

63 Sun X Q, Zhang J, Nie Q. Inferring latent temporal progression and regulatory networks from cross-sectional transcriptomic data of cancer samples. PLoS Comput Biol 2021; 17(3): e1008379

64 Sun X Q, Zhang J, Zhao Q. . Stochastic modeling suggests that noise reduces differentiation efficiency by inducing a heterogeneous drug response in glioma differentiation therapy. BMC Syst Biol 2016; 10(1): 73

65 Wang Z H, Zhang L, Sagotsky J. . Simulating non-small cell lung cancer with a multiscale agent-based model. Theor Biol Med Model 2007; 4: 50

66 Zhang J, Guan M G, Wang Q L. . Single-cell transcriptome-based multilayer network biomarker for predicting prognosis and therapeutic response of gliomas. Briefings Bioinf 2020; 21(3): 1080–1097

67 Zhang J J, Zhu W B, Wang Q L. . Differential regulatory network-based quantification and prioritization of key genes underlying cancer drug resistance based on time-course RNA-seq data. PLoS Comput Biol 2019; 15(11): e1007435

68 Zhang L, Athale C A, Deisboeck T S. Development of a three-dimensional multiscale agent-based tumor model: simulating gene-protein interaction profiles, cell phenotypes and multicellular patterns in brain cancer. J Theor Biol 2007; 244(1): 96–107

69 Zhang L, Chen L L, Deisboeck T S. Multi-scale, multi-resolution brain cancer modeling. Mathe Comp Simul 2009; 79(7): 2021–2035

70 Zhang L, Wang Z H, Sagotsky J A. . Multiscale agent-based cancer modeling. J Math Biol 2009; 58(4/5): 545–559

71 Zheng X, Wise S M, Cristini V. Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull Math Biol 2005; 67(2): 211–259

72 Zheng Y J, Bao J G, Zhao Q Y. . A spatio-temporal model of macrophage-mediated drug resistance in glioma immunotherapy. Mol Cancer Ther 2018; 17(4): 814–824