Mathematical Analysis of a New Model of Bone Pattern Formation
Total Page：16
File Type：pdf, Size：1020Kb
INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH POLISH ACADEMY OF SCIENCES PHD DISSERTATION Mathematical analysis of a new model of bone pattern formation Paramita Chatterjee Supervisor Bogdan Ka´zmierczak, Institute Professor Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw, Poland 2019 Abstract In this work we study a mathematical model proposed in reference [16] to describe processes of pattern formation in morphogenesis. In fact, the model is dedicated to bone formation phenomena in vertebrate embryos. The first equation of the considered system does not fall into any of the three basic classes of partial differential equations. To be more precise, this equation is parabolic in time and space and hyperbolic with respect to time and a pair of auxiliary independent variables describing the state of the cells. This fact does not allow us to use, at least straightforwardly, the usual methods of analysis assigned either to strictly parabolic or strictly hyperbolic problems and, according to our knowledge based on literature search and private communications, there are no theorems guaranteeing the existence of such equations even the homogeneous case. Similar difficulties occur when we attempt to carry out the numerical simulations of the model. Our study of the problems connected with the analysed system is divided into two parts. In first part, see Part II, we consider a scalar equation retaining the basic difficulties of the system. We do not take into account the non-local terms (which are the source of aggregation phenomena), but concentrate on the existence of solutions to linear equations, homogeneous as well as inhomogenous ones. We manage to construct the solutions by means of appropriately defined solution kernels, both in the spatially unbounded as well as bounded case. We prove that the constructed solutions are unique in appropriate spaces of functions. We can also show the validity of the expressions defining solutions to homogeneous equations, when the initial data are given in the product form and the problem can be solved starightforwardly. In second part, see Part III, we deal with the whole system of three equations describing the analysed system, however we use another approach to prove the existence of its solutions. The approach consists in assigning to this system a modified version of the Rothe numerical scheme with time interval discretized into intervals of the lenght ∆t. By deriving a series of a priori estimates, we are able to prove that the proposed numerical scheme produces, in the limit ∆t ! 0, a solution to the system, in which, similarly to Part II we replace the non-local term by local functions. iii Acknowledgments I would like to express my gratitude to my supervisor Bogdan Ka´zmierczak,for his deliberate discussions, continuous support, care, protection and indulgence that I received from him during my PhD study and research. His patience, motivation and exceptionally refined intuition has always been a great source of inspiration to me. I also would like to thank all the Professors, my colleagues and the staffs of my Institute. A special thanks to Slawomir Bialecki for his constant help. I also thanks to Dr. Marek Kocha´nczykfor his help. My research was partially sponsored by the grants 2016/21/B/ST1/03071. I would like to thank the principal investigator of the grant Bogdan Ka´zmierczak,for the financial help. v Contents Part I Introduction.........................1 1 Background and motivation .................................. 1 2 Specificity of the system and the objective of the dissertation ............ 4 Part II Linearized scalar equation representing system (1.11){(1.13). The Green's function approach...................6 3 The case of γ~ and δ~ independent of x and t ....................... 6 4 Uniqueness of solutions .................................... 20 5 Asymptotics of the solution given by Lemmas 3.8 and 3.9 .............. 23 5.1 Asymptotic weak limit of the terms (@=@T1)Γ(T1)R and (@=@T8)B(T8)R with R given by Lemma 3.8 ....................................... 24 5.2 Asymptotic weak limit of the terms (@=@T1)Γ(T1)R and (@=@T8)B(T8)R with R given by Lemma 3.9........................................ 26 6 Weak formulation of Eq.(3.1) ................................. 28 7 Integral equality satisfied by the function given by (3.52) ............... 30 8 The case of the product initial data ............................. 31 9 Extension to the equation with added diffusion terms ................. 36 10 Remarks on the existence of the Green's function for the Neumann problems in bounded regions ......................................... 42 11 The case of bounded regions ................................. 44 12 Final remarks ........................................... 45 12.1 Generalization to bigger dimensions............................ 45 12.2 Convolution notation.................................... 46 Part III Existence theorems via the Rothe method......... 48 13 Modified discrete Rothe method ............................... 48 14 Preliminary lemmas and properties ............................. 49 14.1 Preliminary lemmas .................................... 49 14.1.1 The uniqueness of solutions ............................ 50 15 Discrete-continuous numerical scheme for the simplified system .......... 51 15.1 Main assumptions...................................... 52 15.2 Description of the method of proving the existence of solutions to a variation of system (1.11)-(1.13)......................................... 54 15.3 Modification of the function γ ............................... 56 15.4 Properties of the function δ and the modified function γ∗ . 57 15.5 Estimate of the upper bound of the functions Ri .................... 59 15.6 Bounds for the evolution of the support of the function Ri . 60 u;i u;i 15.7 Estimates of the a priori bounds of the functions c1 and c8 . 61 i 15.8 Estimates of first order derivatives of R with respect to Tm . 65 15.9 Estimates of the first order derivatives of Ri with respect to the components of x inside Ω ........................................... 67 vii 15.10 Estimates of the first order derivatives of Ri with respect to the components of x at @Ω 71 i 15.11 Second order derivatives of R with respect to Tl and Tm . 72 i 15.12 Third order derivatives of R with respect to Tl, Tm and Tp . 74 i 15.13 Mixed second order derivatives of R with respect to xk and Tm . 74 i 15.14 Mixed third order derivatives R with respect to xk, Tm and Tl . 76 15.15 Basic lemma concerning the difference between functions corresponding to subsequent values of i .......................................... 76 1+β i 15.16 Estimates of Cx norms of the functions R ...................... 86 u;i 15.17 Estimates of the higher order norms of the functions ck . 87 i 15.18 Estimates of first order derivatives of Z with respect to xk . 87 2+β i 15.19 Estimates of Cx norms of the functions R ...................... 96 15.20 Estimate of differences Zi − Zi−1 ............................. 97 16 Convergence of the sequences as ∆t ! 0.......................... 98 17 Conclusions ............................................ 102 References ............................................... 104 A Appendix A { Laplace operator in IRm in local coordinates connected with an (n − 1) dimensional hypersurface ............................... 106 B Appendix B { Mathematical modeling of chondrogenic pattern formation during limb development: Recent advances in continuous models .............. 109 viii Part I Introduction 1 Background and motivation Morphogenesis is one of the most interesting phenomenon in biology. It is extremely intriguing to explain, how from an initially homogeneous set of identical cells, spatial patterns composed of differ- entiated cells, leading to the formation of tissues organs and finally whole organisms can be created. One of the important example of morphogenetic processes is the vertebrate limb formation. The formation of the skeletal pattern in vertebrate limbs received particular attention by the researchers. To be more precise, the mechanism of cellular and molecular interactions during the growth of the avian forelimb, for example, spatio-temporal differentiation of cartilage, such that the number of bone primordial changes in time from one (humerus), to two (radius and ulna) and to three (digits) get significant attention. Here we should keep in mind that the mechanism of chondrogenesis may differ from species to species, but its main features are common to all the vertebrates. In the experimental context the bone formation process is most often described for mice or chickens. At the initial stages of embryo development, limb mesenchymal cells started to condense to so called precartilage. After then the precartilage mesenchymal cells organize themselves into spot- or rod-like condensations of nearly uniform size [8,7, 22], which are then turn into definite cartilage, followed by bone. These phenomena, together with appropriate geometry of the limb bud have been the subjects of many biological as well as mathematical models (see e.g. the reviews [33],[27], [28]). In Appendix B (after the References), we attach a review by P. Chatterjee, T. Glimm & B. Kazmierczak [5], submitted to the journal of Mathematical Biosciences, which, among others, relates the considered model to other models of pattern formation, especially to the one presented in [2]. (However, let us emphasize that [5] is not a part of this dissertation and has been appended only for completeness.) The