Tutorials in mathematical biosciences III. Cell cycle, prolifereation, and cancer.
The lectures in this volume are devoted to the cell division cycle, tumor growth and cancer chemotherapy. The first chapter discusses the Novak-Tyson model of the cell division cycle. The cell mass \(M\) is governed by \(dM/dt=\mu M\). The mitosis promoting factor \([MPF]\) is given by \([MPF]=[Cdc13_T]^{-1}(([Cdc13_T]-[preMPF])([Cdc13_T]-[Trimer]))\). The mass is coupled with \([Cdc13_T]\) (a B-type cyclin) given by \[ d[Cdc13_T]/dt=k_1M-k_2^{\prime}[Cdc13_T]-k_2^{\prime\prime} [Ste9][Cdc13_T]-k_2^{\prime\prime\prime}[Slp1][Cdc13_T]. \] The \([preMPF]\) is the total of the tyrosine-phosphorylated Cdc13/Cdc2 with or without Rum1 (a CDK-cyclin dependent kinase-inhibitor): \[ \begin{multlined} d[preMPF]/dt=k_{wee}([Cdc13_T]-[preMPF])-k_{25}[preMPF]+\\-k_2^{\prime} [preMPF] -k_2^{\prime\prime}[Ste9][preMPF]-k_2^{\prime\prime\prime}[Slp1][preMPF].\end{multlined} \] The antagonism between Cdc13/Cdc2 and APC (anaphase promoting complex) is described by the auxiliary factors Ste9 and Slp1 (its concentration is \([Slp1_T]\)): \[ d[Slp1_T]/dt=k_5^{\prime}+k_5^{\prime\prime}[MPF]^4/(J_5^4+[MPF]^4)-k_6[Slp1_T], \] \[ \begin{multlined} d[Slp1]/dt=k_7[IEP]([Slp1_T]-[Slp1])/(J_7+([Slp1_T]-[Slp1]))+\\ -k_8[Slp1]/(J_8+[Slp1])-k_6[Slp1], \end{multlined} \] \[ d[IEP]/dt=k_9[MPF](1-[IEP])/(J_9+(1-[IEP]))-k_{10}[IEP]/(J_{10} +[IEP]), \] \[ \begin{multlined} d[Ste9]/dt=(k_3^{\prime}+k_3^{\prime\prime}[Slp1])(1-[Ste9])/ (J_3+(1-[Ste9]))+\\-((k_4^{\prime}[SK]+k_4[MPF])[Ste9])/(J_4+[Ste9]). \end{multlined} \] The Rum1 concentration is given by \[ d[Rum1_T]/dt=k_{11}-(k_{12}+k_{12}^{\prime}[SK]) +k_{12}^{\prime\prime} [MPF])[Rum1_T]. \] The Cdc13/Cdc2 inhibits the SK (starter kinase) activity: \[ d[SK]/dt=k_{13}[TF]-k_{14}[SK], \] where \([TF]\) is the transcription factor. Several auxiliary equations, given by means of the Goldbeter-Koshland function, are also needed. The phase-plane and bifurcation analysis of the system is performed with the aim of getting some useful information for a study of the more complex regulatory network of the mammalian cell cycle. A kinetic model of coupling between the cell cycle and the apoptosis is also analyzed. It implements ultrasensitive responses of the signaling, cell cycle and apoptosis modules: for \(\Sigma=\) signaling molecule, \(S_2= \) active signaling protein, \(G_2=\) control node of transcription factors, \(C_2= \) active cell cycle marker, \(A_2=\) active apoptosis marker and \(S_1\), \(C_1\), \(A_1\) the corresponding inactive forms of the molecules, the equations read as \[ d\Sigma/dt=\varepsilon^0(k_s-k_{sd1}\Sigma-k_{sd2}C_2\Sigma), \] \[ dS_2/dt=\varepsilon^{-2}(k_1\Sigma S_1(K_{M1}+S_1)^{-1}-v_{m1}S_2(K_{Mr1}+S_2)^{-1}), \] \[ dG_2/dt=\varepsilon^{-3}(k_2S_2+k_{2a}C_2-k_{m2}G_2-k_{m2a}C_2G_2), \] \[ dC_2/dt=\varepsilon^{-1}(k_3G_2C_1(K_{M3}+C_1)^{-1}-v_{m3}C_2(K_{Mr3} +C_2)^{-1}), \] \[ dA_2/dt=\varepsilon^2(k_4G_2A_1(K_{M4}+A_1)^{-1}+k_{4a}C_2A_1(K_{M4} +A_1)^{-1}-v_{m4}A_2(K_{Mr4}+A_2)^{-1}). \] The second chapter discusses angiogenesis from a biochemical/mathematical point of view. After a thorough presentation of the anatomy and biochemistry involved (see also the most helpful dictionary in the end of the chapter), several mathematical models are discussed: the pioneering Keller-Segal model, the ODE system of enzyme kinetics that takes into account the Michaelis-Menten hypothesis, other recent models of chemosensitive movements. The third chapter investigates in vivo cancer cell invasion of tissues. An excursus of the large medical and mathematical literature devoted to the topic precedes the numerical analysis of the next system of equations: the rate of change of cell density taking into account random motility, chemotaxis, haptotaxis, cell proliferation is given by \[ \partial c/\partial x=D_c(\partial^2 c/\partial^2 x)-(\partial/\partial x)[\chi_c c(\partial u/\partial x)+\zeta_c c(\partial p/\partial x)+\xi_c c(\partial v/\partial x)]+\Phi_{13}cu+\mu_1 c(1-cc_0^{-1}), \] the rate of change of ECM density taking into account plasmin formation, PAI-1/uPA binding and PAI-1 inhibition reads as \[ \partial v/\partial t=(-\delta vm)+\Phi_{21}up-\Phi_{22}vp+\mu_2 v(1-vv_0^{-1}), \] the rate of change of uPA concentration taking into account the diffusion, the cancer cells and their binding and the PAI-1 inhibition reads as \[ \partial u/\partial t=D_u(\partial^2u/\partial x^2)-\Phi_{31}pu-\Phi_{33}cu+\alpha_{31}c, \] the rate of change of PAI-1 concentration taking into account the diffusion, the plasmin activation/cell secretion, the VN and uPA binding is given by \[ \partial p/\partial t=D_p(\partial^2p/\partial x^2)-\Phi_{41}pu-\Phi_{42}pv+\alpha_{41}m, \] the rate of change of plasmin concentration taking into account the diffusion, the cells, the uPA/PAI-1 and PAI-1/VN binding reads as \[ \partial m/\partial t=D_m(\partial^2m/\partial x^2)-\Phi_{51}pu+\Phi_{52}pv+\Phi_{53}uc. \] The fourth chapter analyzes a model of the spatial-temporal response of cytotoxic T-lympho\-cytes to a solid tumor: for \(E\), \(T\), \(C\), \(E^*\), \(T^*\) and \(\alpha\) standing for the local densities/concentrations of TICLs (tumor-infiltrating cytotoxic lymphocytes), tumor cells, TICL-tumor cell complexes, inactivated TICLs, lethally hit tumor cells and a generic chemokine the differential system reads as \[ \partial E/\partial t=D_1\Delta E-\chi\nabla\cdot(E\nabla\alpha)+sh(x)+fC(g+T)^{-1}-d_1E-k_1ET+(k_{-1}+k_2p)C \] (\(h\) is the Heaviside function modeling the existence of a subregion of the domain of interest where lymphocytes do not reside and which is penetrated by effector cells through the process of diffusion and chemotaxis only), \(\partial\alpha/\partial t=D_2\Delta\alpha+k_3C-d_4\alpha\), \[ \partial T/\partial t=D_3\Delta T+b_1(1-b_2T)T-k_1ET+(k_{-1}+k_2(1-p))C, \] \[ \partial C/\partial t=k_1ET-(k_{-1}+k_2)C,\;\partial E^*/\partial t=k_2(1-p)C-d_2E^*,\;\partial T^*/\partial t=k_2pC-d_3T^*. \] Numerical, bifurcation and traveling-waves investigations are presented and commented upon. Among other novelties, the system allows for a Hopf bifurcation which is realistic because of the cytotoxic activity of the TICLs. The fifth chapter discusses a control/system theory approach (Pontryagin's maximum principle, higher order Legendre-Clebsh condition, Nyquist theorem) to cancer chemotherapy with an emphasis of evolving drug resistance caused e.g. by gene amplification. The last chapter presents several results about tumor growth from the point of view of a free boundary problem: if \(\Omega(t)\) is the tumor region of boundary \(\Gamma(t)\), \(c\) is the concentration of nutrients, \(K_D\) is the death rate (the nutrients become necrotic or go into apoptosis), the proliferating rate is \(K_P\), the proliferating cells (\(p=\)proliferating cells) become quiescent (\(q=\)quiescent cells) at a rate \(K_Q\) and their death rate is \(K_A\), the proliferation rate is \(K_B\), the rate of dead (removed) cells (\(n=\)dead cells) is \(K_R\) then \[ \varepsilon_0(\partial c/\partial t)=D_c\Delta c+\Gamma(c_B-c)-\lambda c \] (\(c_B\) is the nutrient concentration in the vasculature, \(\Gamma\) is the rate of blood/tissue transfer), \[ \partial p/\partial t +\text{div}(pv)=[K_B(c)-K_Q(c)-K_A(c)]p+K_P(c)q, \] \[ \partial/\partial t+\text{div}(qv)=-[K_P(c)+K_D(c)]q+K_Q(c)p, \] \[ \partial n/\partial t+\text{div}(nv)=K_D(c)q+K_A(c)p-K_Rn \text{in}\;\Omega(t), \] \(\Delta\sigma=K_R-[K_B(c)+K_R]p-K_Rq\), where \(t>0\), the velocity \(v\) of fluid flow obeys Darcy's law \(v=-\nabla\sigma\) and \(p+q+n=1=\text{constant}\). The boundary conditions read as: \(c=\overline(c)\), \(\sigma=\gamma k\) (\(\gamma=\)surface tension coefficient, \(k=\)mean curvature), \(\partial\sigma/\partial n=-V_n\) (\(V_n=\)velocity of the free boundary) on \(\Gamma(t)\) for \(t>0\).
scientific article; zbMATH DE number 5016782