Evolutionary dynamics of cancer: from epigenetic regulation to cell population dynamics -- mathematical model framework, applications, and open problems
EEVOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETICREGULATION TO CELL POPULATIONDYNAMICS—MATHEMATICAL MODEL FRAMEWORK,APPLICATIONS, AND OPEN PROBLEMS
JINZHI LEI
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, China.
Abstract
Predictive modeling of the evolutionary dynamics of cancer is a challengeissue in computational cancer biology. In this paper, we propose a general mathematicalmodel framework for the evolutionary dynamics of cancer with plasticity and heterogeneityin cancer cells. Cancer is a group of diseases involving abnormal cell growth, during whichabnormal regulations in stem cell regeneration are essential for the dynamics of cancerdevelopment. In general, the dynamics of stem cell regeneration can be simplified as aG phase cell cycle model, which lead to a delay differentiation equation. When cell het-erogeneity and plasticity are considered, we establish a differential-integral equation basedon the random transition of epigenetic states of stem cells during cell division. The pro-posed model highlights cell heterogeneity and plasticity, and connects the heterogeneitywith cell-to-cell variance in cellular behaviors, e.g. , proliferation, apoptosis, and differen-tiation/senescence, and can be extended to include gene mutation-induced tumor devel-opment. Hybrid computations models are developed based on the mathematical modelframework, and are applied to the process of inflammation-induced tumorigenesis and tu-mor relapse after CAR-T therapy. Finally, we give rise to several mathematical problemsrelated to the proposed differential-integral equation. Answers to these problems are crucialfor the understanding of the evolutionary dynamics of cancer. Key words : stem cell regeneration; differential-integral equation; cancer development;computational cancer biology; open problems1.
Introduction
According to the World Health Organization (WHO), cancer has become the first rank-ing cause of death at ages below 70 years in 48 countries at 2015. Cancer has been aserious issue for public health. How cancers arise from normal tissues? Why is cancers sohard to treat? Despite many years researches, people know little about the evolutionarydynamics of cancer. Multidisciplinary study is required to uncover the mechanism of can-cer development. Recent years, mathematical oncology–the use of mathematics in cancerresearch–has gained momentum with the rapid accumulation of data and applications ofmathematical methodologies[33]. a r X i v : . [ q - b i o . CB ] A ug VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS2
Cancer is a group of diseases involving abnormal cell growth with the potential to invadeor spread to other parts of the body. Hence, abnormal stem cell regeneration is essentialin cancer development. Many mathematical models of the population dynamics have beenwidely studied in understanding how stem cell regeneration is modulated in different con-text [2, 7, 17, 20, 25, 26, 28, 29, 34, 43, 47]. In most models, the dynamics of a homogeneouscell pool or the lineage of several homogeneous subpopulations were formulated through aset of differential equations. Novel experimental techniques developed in recent years allowus to investigate the heterogeneity of cells at single cell level [4, 8, 10, 16, 37, 45]. Theheterogeneity is mostly originated from random changes of epigenetic state at each cellcycle, and often induce cell plasticity and diversity which form the main sources of drugresistance in cancer therapy[11, 21, 40, 32]. Based on the stochastic transition of epigeneticstate during cell division, mathematical models for the heterogeneous population dynam-ics can be formulated as discrete dynamical system or continuous differentiation-integralequations [23, 38]. Moreover, hybrid models are often developed to simulate the processof tumor development and treatment of tumors as multicellular heterogeneous system oftumors[3].In this paper, we present a general mathematical framework of modeling the evolution-ary dynamics of cancer with plasticity and heterogeneity in the cells. We first introduce adelay differential equation obtained from an age-structure model of G phase stem cell re-generation, and next propose the mathematical formulations for heterogeneous populationdynamics with consideration of epigenetic state transition during cell divisions. Next, weintroduce application of the proposed framework in case studied of cancer development.Finally, we give rise mathematical problems related to the proposed mathematical model.Answers to these problems are crucial for understanding the evolutionary dynamics ofcancer. 2. A simple model– G cell cycle model First, we introduce a simple model of stem cell regeneration–the G cell cycle modelestablished by Burns and Tannock in 1970 [2]. This model assumed a resting phase (G )between two cell cycles. Stem cells at cell cycling are classified into resting or proliferatingphases (Figure 1). During each cell cycle, a cell in the proliferating phase either undergoesapoptosis or divides into two daughter cells; however, a cell in the resting phase eitherirreversibly differentiates into a terminally differentiated cell or returns to the proliferatingphase. Resting phase cells can also be removed from the stem cell pool due to cell deathor senescence.Mathematically, the above process can be modeled by an age-structure model for cellnumbers in the resting phase and proliferating phase. Let s ( t, a ) for the number of stemcells at the proliferating phase, the age a = 0 is the time of entry into the proliferativestate, and Q ( t ) the number of resting-phase stem cells. The above assumptions yield the VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS3 G0 κµ proliferating phase apoptosis differentiation, death, senescence proliferation: β ( Q ) τ Figure 1.
The G model of stem cell regeneration. During stem cell re-generation, cells in the resting phase either enter the proliferating phasewith a rate β , or be removed from the resting pool with a rate κ due todifferentiation, aging, or death. The proliferating cells undergo apoptosiswith a probability µ .following partial differential equations[2](1) ∇ s ( t, a ) = − µs ( t, a ) , ( t > , < a < τ ) dQdt = 2 s ( t, τ ) − ( β ( Q ) + κ ) Q, ( t > . Here ∇ = ∂/∂t + ∂/∂a . The coefficient 2 means that each mother cell generates twodaughter cells after cell division. The boundary condition at a = 0 is as follows:(2) s ( t,
0) = β ( Q ( t )) Q ( t ) , and the initial conditions are(3) s (0 , a ) = g ( a ) , (0 ≤ a ≤ τ ); Q (0) = Q . Equations (1)-(3) give a general age-structure model of homogeneous stem cell regeneration.Integrating (1)-(2) through the method of characteristic line, and consider the long-termbehavior ( t > τ ), we obtain a delay differential equation model [24](4) dQdt = − ( β ( Q ) + κ ) Q + 2 e − µτ β ( Q τ ) Q τ , where Q τ ( t ) = Q ( t − τ ). This equation describes the general population dynamics of stemcell regeneration.The proliferation rate β ( Q ) describes how cells regulate the self-renewal of stem cellsthrough secreted cytokines, and is often given by a decrease function of form [1, 25](5) β ( Q ) = β θ n θ n + Q n + β . Here, a non-zero constant β is included to represent the self-sustained growth signals ofcancer cells[14]. VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS4
When β ( Q ) is given by (5), the equation (4) has a unique positive steady state if andonly if(6) β + β > η > β , here η = κ e − µτ − . The steady state is given by(7) Q ∗ = η (cid:20) β + β − ηη − β (cid:21) /n . Therefore, abnormal growth ( Q ∗ → ∞ ) may occur when η → β , i.e. , decreasing η (bydecreasing κ or µ ) or increasing β . Hence, the simple G cell cycle model (4) implies threepossible conditions for abnormal cell growth: dysregulation in the pathways of differenti-ation and/or senescence (decreasing κ ), sustaining proliferative signaling (increasing β ),and evasion of apoptosis (decreasing µ ). These conditions are well known hallmarks ofcancer [14]. 3. A model with cell heterogeneity and plasticity
Modele equation.
The above G cell cycle model describes the population dynamicsof homogeneous stem cell regeneration. However, cells are different by their epigeneticstates, including the patterns of DNA methylations, nucleosome histone modifications,and the transcriptomics. These epigenetic states can change from mother cell to daughtercells during cell cycling through the re-distribution of modification markers, nucleosomeassembling, or stochastic molecular partitions (Figure 2) [31, 35, 36, 37, 41, 44]. κµ apoptosis differentiation, death, senescence proliferation: β ( Q ) Figure 2.
Illustration of the model for heterogeneous stem cell regenera-tion. Similar to Figure 1. The heterogeneity is represented by cells withdifferent colors, and the hence the rate functions are dependent on the epi-genetic state of the cell.To model the heterogeneity in stem cells, we introduce a valuable x (often a high dimen-sional vector) for the epigenetic state of cells, and Ω for the space of all possible epigeneticstates in resting phase stem cells. The epigenetic state x represents intrinsic cellular statesthat may change during cell division. Let Q ( t, x ) the number of stem cells at time t in the VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS5 resting phase with epigenetic state x , the total cell number is given by(8) Q ( t ) = (cid:90) Ω Q ( t, x ) d x . Proliferation of each cell is regulated by the signaling pathways that are dependent onextracellular cytokines released by all cells in the niche and the epigenetic state x of thecell[1, 20, 29]. Hence, the proliferation rate β has a form β ( ˆ Q, x ), where ˆ Q is the effectiveconcentration of growth inhibition cytokines given by(9) ˆ Q ( t ) = (cid:90) Ω Q ( t, x ) ζ ( x ) d x , here ζ ( x ) is the rate of cytokines secretion. Moreover, the apoptosis rate µ , the cell cycleduration τ , and the differentiation rate κ are dependent on the epigenetic state x , and aredenoted as µ ( x ), τ ( x ), and κ ( x ), respectively.Moreover, to consider cell plasticity at each cell cycle, we introduce a transition function p ( x , y ) for the inheritance probability, which represent the probability that a daughter cellof state x comes from a mother cell of state y . It is obvious that (cid:90) Ω p ( x , y ) d x = 1for any y ∈ Ω.Now, similar to (1), when stem cell heterogeneity is include, we obtain the correspondingage-structure model equation(10) ∇ s ( t, a, x ) = − µ ( x ) s ( t, a, x ) , ( t > , < a < τ ( x )) ∂Q ( t, x ) ∂t = 2 (cid:90) Ω s ( t, τ ( y ) , y ) p ( x , y ) d y − ( β ( ˆ Q ( t ) , x ) + κ ( x )) Q ( t, x ) , ( t > . and s ( t, , x ) = β ( ˆ Q ( t ) , x ) Q ( t, x ) , ˆ Q ( t ) = (cid:90) Ω Q ( t, x ) ζ ( x ) d x . Here we note that x can be considered as a parameter for the first equation, hence, we canapply the characteristic line method and have s ( t, τ ( x ) , x ) = β ( ˆ Q ( t − τ ( x )) , x ) Q ( t − τ ( x ) , x ) e − µ ( x ) τ ( x ) . Thus, substituting s ( t, τ ( x ) , x ) into the second equation in (10), we obtain the followingdelay differential-integral equation(11) ∂Q ( t, x ) ∂t = − Q ( t, x )( β ( ˆ Q, x ) + κ ( x ))+ 2 (cid:90) Ω β ( ˆ Q τ ( y ) , y ) Q ( t − τ ( y ) , y ) e − µ ( y ) τ ( y ) p ( x , y ) d y , ˆ Q ( t ) = (cid:90) Ω Q ( t, x ) ζ ( x ) d x , Here ˆ Q τ = ˆ Q ( t − τ ). VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS6
The equation (11) gives the basic dynamical equation of heterogenous stem cell regen-eration with epigenetic transition. Biologically, the equation (11) connects different scalecomponents(Figure. 3): the gene expressions at single cell level ( x ), the population dy-namic properties ( β ( ˆ Q, x ) , κ ( x ), and µ ( x )), cell cycle ( τ ( x )), and epigenetic modificationinheritance ( p ( x , y )). Thus, this equation provides a framework of mathematical model forstem cell regeneration, and can be applied to different problems related to cell regenera-tion, such as development, aging, tumor development, etc . For detail discussions of (11),referred to [22]. Figure 3.
Framework of the the mathematical model of heterogeneousstem cell regeneration.3.2.
The transition function p ( x , y ) . In the above equation, the transition function p ( x , y ) is important to connect cell heterogeneity with plasticity. However, biologically wecannot measure this function directly, and do not know the possible form neither. Here,we propose a possible form of the transition function through numerical simulation basedon a computational model of the inheritance of histone modification[18, 19].Let the epigenetic state x (0 ≤ x ≤
1) represent the fraction of nucleosomes (in asegment of DNA sequences), and p ( x, y ) the probability of a daughter cell with state x given the mother cell of state y . Simulations based on a computation model of histonemodification/transition suggest that the probability p ( x, y ) can be approximately describedby a conditional Beta-distribution p ( x, y ) = x a ( y ) − (1 − x ) b ( y ) − B ( a ( y ) , b ( y )) , B ( a, b ) = Γ( a )Γ( b )Γ( a + b ) . Here, the Beta-distribution depends on two shape parameters a and b , which are in turndependent on the state y of the mother cell. Hence, the Beta-distribution can be quitegeneral with different definitions of the functions a ( y ) and b ( y ). VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS7
In general, when the epigenetic state x = ( x , · · · , x n ) that includes multiple variables,we can extend the above Beta-distribution by the multiply rule p ( x , y ) = n (cid:88) i =1 p i ( x i , y ) , where p i ( x i , y ) = x a i ( y ) − i (1 − x i ) b i ( y ) − B ( a i ( y ) , b i ( y )) . To determine the functions a i ( y ) and b i ( y ), if we write the mean and variance of x i , giventhe state y , as E( x i | y ) = φ i ( y ) , Var( x i | y ) = 11 + η i ( y ) φ i ( y )(1 − φ i ( y )) , through predefined function φ i ( y ) and η ( y ), the shape parameters are given by a i ( y ) = η i ( y ) φ i ( y ) , b i ( y ) = η i ( y )(1 − φ i ( y )) . Here, the functions φ i ( y ) and η i ( y ) always satisfy0 < φ i ( y ) < , η i ( y ) > . As an example, if we consider a situation of one epigenetic state x (0 ≤ x ≤
1) whichis analogous to the stemness of a cell [27]. The stemness may affect cell proliferation anddifferentiations so that the rates β and κ are dependent on the epigenetic state x . Moreover,we assume that ξ ( x ) ≡ Q ( t ) = (cid:90) Q ( t, x ) dx. In this case, we have a one dimensional differential-integral equation(13) ∂Q ( t, x ) ∂t = − Q ( t, x )( β ( ˆ Q ( t ) , x ) + κ ( x )) + 2 e − µτ (cid:90) β ( ˆ Q ( t − τ ) , y ) Q ( t − τ, y ) p ( x, y ) dy. The inheritance probability function p ( x, y ) is defined as(14) p ( x, y ) = x a ( y ) (1 − x ) b ( y ) B ( a ( y ) , b ( y )) , a ( y ) = η ( y ) φ ( y ) , b ( y ) = η ( y )(1 − φ ( y ))with pre-defined functions φ ( y ) and η ( y ).Specifically, we can define [22](15) β ( ˆ Q, x ) = ¯ β × a x + ( a x ) a x ) θθ + ˆ Q + β , and(16) κ ( x ) = κ ×
11 + ( b x ) . VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS8
The functions β and κ defined by (15) and (16) are taken so that a cell has low proliferationrate if the stemness if either high or low, and large proliferation rate if the stemness isintermediate, and the differentiation rate decreases with the stemness.3.3. Model cancer development with genetic heterogeneity.
In the above equa-tions, we only consider heterogeneous in epigenetic states. However, gene mutations areimportant for cancer development, which form intratumoral heterogeneity of cancer cells.The cells may have diverse combinations of mutations. To model the genetic heterogeneity,we extend (11) to include mutant types. Assuming that there are m genetic types, we let Q i ( t, x ) the cell population with genetic type i (1 ≤ i ≤ m ) and epigenetic state x , and p i,j the mutation rate from genetic type i to type j . Moreover, we assume that mutations onlyhappen during cell division (DNA replication). Thus, when gene mutations are include,the equation (11) is rewritten as ∂Q i ( t, x ) ∂t = − Q i ( t, x )( β i ( ˆ Q, x ) + κ i ( x ))(17) + 2(1 − (cid:88) j (cid:54) = i p i,j ) (cid:90) Ω β i ( ˆ Q τ i ( y ) , y ) Q i ( t − τ ( y ) , y ) e − µ i ( y ) τ i ( y ) p ( x , y ) d y + 2 (cid:88) j (cid:54) = i p j,i (cid:90) Ω β j ( ˆ Q τ ( y ) , y ) Q j ( t − τ ( y ) , y ) e − µ j ( y ) τ j ( y ) p ( x , y ) d y , (1 ≤ i ≤ m )where ˆ Q ( t ) = m (cid:88) i =1 (cid:90) Ω Q i ( t, x ) ξ ( x ) d x . Equation (17) gives a general equation to include gene mutations. Given the mutanttype combination network p i,j , this equation describes the evolutionary dynamics inducedby gene mutations. Mathematically, the mutation may affect the cell behavior parameters,so that the rate constants β, κ, µ , and τ are dependent on the mutant types.4. Applications to the dynamics of cancer development
Hybrid computational model of multicellular tissues.
The equations (11) and(17) provide general mathematical frameworks to model stem cell regeneration when het-erogeneity and plasticity in epigenetic or genetic states are included. These frameworkscan be used to described many biological processes associated with stem cell regeneration,including development, aging, and cancer evolution. Nevertheless, many mathematicalproblems associated with the model remain unsolved (to be detailed latter). Both equa-tions (11) and (17) include integrals over all epigenetic states, and it is expansive to solvethe equations numerically when high dimensional epigenetic states are considered. Thus, inapplications, we often develop hybrid computational models for multicellular tissues basedon the above frameworks. Hybrid models are often used to simulate tumor development
VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS9 as complex multicellular heterogeneous systems[3]. Classical hybrid models combine dis-crete equations to describe individual cells and continuous equations for microenvironmentfactors or intercellular components. Now, the field of tumor modeling have reached intoother mathematical areas and combined continuous or discrete models with concepts fromfluid dynamics, game theory, machine learning, or optimization methods to provide morepowerful predictive models[3].The above mathematical framework suggest a hybrid model that combines discrete sto-chastic process for the epigenetic/genetic state of individual cells with continues model ofcell population growth (here, we state the numerical scheme for epigenetic state heterogene-ity, and it is similar when genetic heterogeneity is included). In this model, a multicellularsystem is represented by a collection of epigenetic states in each cell as Ω t = (cid:110) [ C i ( x i )] Q ( t ) i =1 (cid:111) (here Q ( t ) represents the number of resting-phase stem cells at time t ). During a time inter-val ( t, t + dt ), each cell ( C i ( x i )) undergoes proliferation, apoptosis, or terminal differentialwith a probability given by the kinetic rates (with probability of proliferation, apoptosis, ordifferentiation given by β ( ˆ Q, x i )) dt, µ ( x i ) dt , or κ ( x i ) dt ). Thus, the probabilities of differentcell behaviors are dependent on the epigenetic state of each cells. The total cell number Q ( t ) changes after a time step dt in accordance with the behaviors of all cells. When a cellundergo proliferation, the epigenetic state of daughter cells change randomly according tothe transition function p ( x , y ). In this hybrid model, all detail molecular interactions arehidden into the kinetics rates and the transition function. Moreover, we can also includestochastic process of differential equations for the signaling dynamics within one cell cycle,as well as microenvironmental variables that may depend on the cell behavior of all cells.The propose hybrid model can be implement by single-cell-based models through GPUarchitecture[39].4.2. Application to inflammation-induced tumorigenesis.
Chronic inflammation isa serous risk factor for many cancers. Infection-driven chronic inflammation is linked toapproximately 15% of the global cancer burden[9, 6, 30]. Cancer risk increases stronglywith the duration and extent of chronic inflammation[12, 5]. However, the routes frominflammation to cancer are poorly understood.In [13], following the above mathematical model framework, we developed a compu-tation model for inflammation-induced tumorigenesis that combines the major processesresponsible for stem cell regeneration, inflammation, and metabolism-immune balance. Inthis model, the population dynamics arise from the dynamics of individual cells, each ofwhich is based on a model of the cell-cycle dynamics, and the proliferation rate is depen-dent on the population size. Each cell is associated with individual genetic heterogeneitydue to DNA damage and pathway mutations. DNA damage may occur to a cell under-going cell division, and trigger the processes of DNA damage repair and cell-cycle arrestor DNA damage-induced apoptosis when the damaged loci are not successively repaired.Non-repaired cells can escape from damage-induced apoptosis and reenter the cell cyclesuch that over time, damage loci accumulate and eventually induce functional gene muta-tions in the specific pathways. Mutant cells in the resting phase cell pool can be cleared by
VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS10 the immune system due to the metabolism-immune balance. The inflammatory microenvi-ronment specifically affects the processes of proliferation, apoptosis, and the DNA damageresponse. Gene mutations were not considered explicitly in the model; however, the mu-tations were identified by their effects on the relevant physiologic processes. Mutations toeight pathways were considered (Table 1), each type mutation occur with a given probabil-ity at each cell cycle. When a mutation occur to a cell, the corresponding parameter valuechanges (either up-regulated or down-regulated) in the cell. Each cell may have differentmutant types.
Table 1.
Pathway mutations considered in the model of inflammation-induce tumorigenesis[13].
Symbol Description ∆Prolif Cell proliferation rate∆FSProlif Feedback strength to cell proliferation∆Diff Cell differentiation rate∆Apop Cell apoptosis rate∆Damage Probability of DNA damage induction∆Repair DNA damage repair efficiency∆Escape Probability of DNA damage-induced cell escape∆MIB Probability of metabolism-immune balanceBased on model simulations, we are able to reproduce the process of inflammation-induced tumorigenesis: from normal to precancerous, and from precancerous to malignanttumors. According to model simulations, starting from serve inflammation, most patientsdevelop to precancerous with insignificant increase in the cell population in 10 years, anda few patients may further develop to malignant tumors with signifiant increase in cellnumber in latter stages. Moreover, further analysis shown that mutations to the fourpathways, proliferation, apoptosis, differentiation, and metabolism-immune balance, arecrucial for cancer development, and there are multiple pathways of tumorigenesis[13].4.3.
Application to tumor relapse after CAR-T therapy.
CAR-T therapy targetingCD19 has been proved to be an effective therapy for B cell acute lymphoblastic leukemia(B-ALL). The majority of patients achieve a complete response following a single infusionof CD19-targeted CAR-T cells; however, many patients suffer relapse after therapy, andthe underlying mechanism remains unclear.In a recent study[46], we applied second-generation CAR-T cells to mice injected withleukemic cells; 60% of the mice relapse within 3 months, and the relapsed tumors retainedCD19 expression but exhibited a profound increase in CD34 transcription. Based on theseobservations, we develop a single-cell based computation model for the heterogeneous re-sponse of the tumor cells to the CAR-T treatment.In the model, we introduced key assumptions that CAR-T induced tumor cells to transi-tion to hematopoietic stem-like cells (by promoted CD34 expression) and myeloid-like cells
VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS11 (my promoted CD123 expression) and hence escape of CAR-T targeting. In the model,each cell was represented by the epigenetic state of marker genes CD19, CD22, CD34,and CD123, which play important roles in the CD19 CAR-T cell response and cell lineagedynamics. The proliferation rate β and differentiation rate κ depend on CD34 expressionlevel through β = β θθ + N × . . . ,κ = κ
11 + (4 . . Here N means the total cell number. The apoptosis rate µ includes a basal rate µ and arate associated with the CAR-T signal µ = µ + µ × Signal , Signal = f ([CD34] , [CD123]) γ [CD19]1 + γ [CD19] + γ [CD22] R ( t ) ,f ([CD34] , [CD123]) = 1(1 + ([CD34] /X ) n )(1 + ([CD123] /X ) n ) . Here R ( t ) is the predefined CAR-T activity. The expression levels of marker genes changedrandomly following the transition probability of Beta-distributions, and the shape param-eters were dependent on state of mother cells and the CAR-T signal. For example, giventhe expression level of CD34 at cycle k as u k , the expression level at cycle k + 1 (denotedby u k +1 ) is a Beta-distribution random number with probability density function p ( u, u k ) = u a (1 − u ) b B ( a, b ) , B ( a, b ) = Γ( a )Γ( b )Γ( a + b ) , with the shape parameters a and b dependent on the average and variance of u k +1 . WhenE( u k +1 ) = φ ( u k ) , Var( u k +1 ) = 11 + m φ ( u k )(1 − φ ( u k )) , then a = mϕ ( u k ) , b = m (1 − ϕ ( u k )) . In the model, we assume the average as φ ( u k ) = 0 .
08 + 1 .
06 ( α u k ) . α u k ) . , and let α = 1 .
45 + 0 . × [CD19] + α × Signalto represent the promotion of CD34 expression by CD19 and the CAR-T signal. For detailsof the model, referred to [46].Model simulations nicely reproduced experimental results, and predicted that CAR-Tcell induced cell plasticity can lead to tumor tumor relapse in B-ALL after CD19 CAR-T treatment. Simulations and mouse experiments further indicated that CD19 positive
VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS12 relapse could be prevented by the combined administration of CD19- and CD123- targetingCAR-t cells administered at specific ratios.5.
Mathematical problems
The proposed mathematical framework (11) is a delay differential-integral equation thatcontains non-local transitions between different epigenetic states. This type equation wasnot seen in most physical problems because of the principle of locality. Mathematically,there are many basic problems remain open for the equation (11). Here, I discuss a fewof them that are basic and important for our understanding of the biological process ofcancer development.For the simplicity, we omit the delay, and consider the equation with only one dimensionepigenetic state. Hence, we have the following equation(18) ∂Q ( t, x ) ∂t = − Q ( t, x )( β ( ˆ Q, x ) + κ ( x )) + 2 (cid:90) Ω β ( ˆ Q, y ) Q ( t, y ) e − µ ( y ) p ( x, y ) dy ˆ Q ( t ) = (cid:90) Ω Q ( t, x ) ζ ( x ) dx. Here x ∈ Ω ⊂ R + , and always assume(19) β ( x ) ≥ , κ ( x ) ≥ , µ ( x ) ≥ , ζ ( x ) ≥ , ≤ p ( x, y ) ≤ , and(20) (cid:90) Ω p ( x, y ) dx = 1 , ∀ y ∈ Ω . Here, we note that by omitting the delay, we do not simply set µ = 0 in the equation(11), but only omit the delays in Q ( t − τ ( y ) , y ) and ˆ Q τ ( y ) , and replace e − µ ( y ) τ ( y ) by e − µ ( y ) with µ ( y ) the apoptosis rate during the proliferating phase.5.1. Question 1: Existence and uniqueness of the steady state solution.
To con-sider the steady state solution of (18), let Q ( t, x ) = Q ( x ) the steady state, then(21) − Q ( x )( β ( ˆ Q, x ) + κ ( x )) + 2 (cid:90) Ω β ( ˆ Q, y ) Q ( y ) e − µ ( y ) p ( x, y ) dy ˆ Q = (cid:90) Ω Q ( x ) ζ ( x ) dx . Substituting ˆ Q into the first equation, Q ( x ) satisfies the eigenvalue problem(22) L ˆ Q [ Q ( x )] = 2 (cid:90) Ω β ( ˆ Q, y ) e − µ ( y ) p ( x, y ) Q ( y ) dy − ( β ( ˆ Q, x ) + κ ( x )) Q ( x ) = 0 . Thus, the problem of existence and uniqueness of the steady state solution is reduced toa problem of finding a positive eigenvalue ˆ Q of the operator L ˆ Q so that the correspondingeigenfunction Q ( x ) is non-negative for all x ∈ Ω. If such eigenvalue ˆ Q exists, the solution of(21) is given by rescaling the corresponding eigenfunction according to the second equationin (21). VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS13
In the case of finite discrete epigenetic state, and when the proliferation rate β is inde-pendent to the epigenetic state, the steady state issue was discussed in [38]. In particular,if β is independent to the epigenetic state, the eigenvalue problem can be rewritten as(23) A [ Q ] = 1 β ( ˆ Q ) Q, where A is a linear operator defined as(24) A [ Q ] = 1 κ ( x ) (cid:20) (cid:90) Ω e − µ ( y ) p ( x, y ) Q ( y ) dy − Q ( x ) (cid:21) . Thus, 1 β ( ˆ Q ) is a positive eigenvalue of the operator A . In the case of finite discrete epige-netic state, the existence for such eigenvalue can be obtained from the Perron-Frobeniustheorem.If the transition probability p ( x, y ) is independent to the state of the mother cell, so that p ( x, y ) = p ( x ), the uniqueness and stability of the steady state were further discussed in[38]. However, it remains open questions for general cases.Biologically, the states of normal and precancerous correspond to stable steady statesunder different conditions. Hence, mathematically identify the existence and stability ofsteady states are important for our understanding of the persistence of different states.5.2. Question 2: Entropy problem.
Let(25) Q ( t ) = (cid:90) Ω Q ( t, x )for the total cell number at time t , and(26) f ( t, x ) = Q ( t, x ) Q ( t )for the fraction of cells with given epigenetic state x . The entropy of the multicellularsystem at time t is defined as(27) E ( t ) = − (cid:90) Ω f ( t, x ) log f ( t, x ) dx. A tedious calculation gives ∂f ( t, x ) ∂t = − f ( t, x ) (cid:90) Ω f ( t, y ) (cid:16) ( β ( ˆ f Q, x ) + κ ( x )) − ( β ( ˆ f Q, y ) + κ ( y )) (cid:17) dy (28) + 2 (cid:90) Ω f ( t, y ) β ( ˆ f Q, y ) e − µ ( y ) ( p ( x, y ) − f ( t, x )) dy,dQdt = Q (cid:90) Ω f ( t, x ) (cid:16) β ( ˆ f Q, x )(2 e − µ ( x ) − − κ ( x ) (cid:17) dx, (29) ˆ f = (cid:90) Ω f ( t, x ) ζ ( x ) dx. (30) VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS14
The derivative of the entropy E ( t ) is given by(31) dEdt = − (cid:90) Ω ∂f ( t, x ) ∂t (1 + log f ( t, x )) dx. Here, we replace ˆ Q with ˆ f Q , and have ˆ f = 1 when ζ ( x ) ≡ epr ) and entropy dissipate rate ( edr ) so that(32) dEdt = epr − edr, and both epr and edr are alway non-negative.Solutions to this problem is essential to answer the question of how the entropy changealong the process of tissue development. In particular, is cancer development a process ofentropy increasing[15, 42]?In the case of homogeneous stem cells, all cells belong to the same epigenetic state, andhence f ( t, x ) = 1, and the entropy E ( t ) ≡
0. Thus, the entropy do not change over thedevelopment process.If the epigenetic state transition is omitted, so that p ( x, y ) = δ ( x − y ), defining(33) γ ( q, x ) = β ( q, x )(2 e − µ ( x ) − − κ ( x )as the net production rate of cells with epigenetic state x , we have(34) ∂f ( t, x ) ∂t = f ( t, x ) γ ( ˆ f Q, x ) − f ( t, x ) (cid:90) Ω f ( t, y ) γ ( ˆ f Q, y ) dy, and(35) dEdt = ∆ E − ∆ E , where(36) ∆ E = − (cid:90) Ω γ ( ˆ f Q, x ) f ( t, x ) log f ( t, x ) dx, ∆ E = E ( t ) (cid:90) Ω γ ( ˆ f Q, x ) f ( t, x ) dx. From (35), for the particular situation that γ ( ˆ f Q, x ) > x ∈ Ω, we have ∆ E > E >
0. In this case, ∆ E gives the entropy production rate, and ∆ E gives theentropy dissipation rate.For the general situation, it is not know how the entropy production rate and dissipationrate should be defined, and under which situation the development process is entropyincreasing.5.3. Question 3: Inverse problem.
In the propose model, the transition probabilityfunction p ( x, y ) is usually not known, and cannot be measured directly from experiments.Hence, it is a challenge issue to estimate this transition function indirectly from experi-mental data.In the equations (28)-(29), for the situation of cancer development, we can estimatethe cell number Q ( t ) (or tumor volume) and the fraction of cells f ( t, x ) through variousmethods, such as imaging, liquid biopsies, biomarker measurement, single-cell sequencing, VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS15 etc . Hence, the inverse problem of obtaining the rate functions β, κ, µ , and ζ , and thetransition function p ( x, y ) is essential for the development of personalized predictive modelfor cancer development. It was proposed that the merger of mechanistic and machinelearning models has became the future of personalization in mathematical oncology[33].In this way, we are trying to merger the propose model with the unwieldy multitude ofdispersed data (imaging, tissue, blood, molecular) to estimate the personalized parameterand to generate optimal clinical decisions for each patient.5.4. Question 4: Local state transition.
In the equations (18) or (18)-(29), the epige-netic transition function p ( x, y ) is global. Here, we assume that the epigenetic state canonly have local transition, i.e.,(37) p ( x, y ) = ϕ ( y − x )so that ϕ ( r ) > | r | < (cid:15) (cid:28)
1. The function ϕ ( r ) satisfies(38) (cid:90) + ∞−∞ ϕ ( r ) dr = 1 , ϕ ( r ) ≥ , ∀ r ∈ R . We further let(39) a = (cid:90) + ∞−∞ rϕ ( r ) dr, D = (cid:90) + ∞−∞ r ϕ ( r ) dr. Substituting (37) into (18), and expand the function ϕ to the second order approximation,we obtain the following close form differential-integral equation ∂f ( t, x ) ∂t = ∂∂x (cid:20)(cid:18) D ∂∂x + 2 a (cid:19) (cid:16) β ( ˆ f Q, x ) e − µ ( x ) f ( t, x ) (cid:17)(cid:21) (40) + f ( t, x ) (cid:18) γ ( ˆ f Q, x ) − (cid:90) f ( t, y ) γ ( ˆ f Q, y ) dy (cid:19) ,dQdt = Q (cid:90) Ω f ( t, x ) γ ( ˆ f Q, x ) dx (41) ˆ f = (cid:90) Ω f ( t, x ) ζ ( x ) dx. (42)From (40)-(42), the above questions of steady state solution, entropy problem, andinverse problem can also be formulated as the problem with local transition. For example,at the steady state, we have (cid:90) Ω f ( t, x ) γ ( ˆ f Q, x ) dx = 0 , and the total number Q ( t ) ≡ Q ∗ . Hence, the steady state solution ( f ( t, x ) = f ( x )) satisfiesa second order differential equation(43) ddx (cid:20)(cid:18) D ddx + 2 a (cid:19) (cid:16) β ( λ, x ) e − µ ( x ) f ( x ) (cid:17)(cid:21) + γ ( λ, x ) f ( x ) = 0 , VOLUTIONARY DYNAMICS OF CANCER: FROM EPIGENETIC REGULATION TO CELL POPULATION DYNAMICS—MATHEMATICAL MODEL FRAMEWORK, APPLICATIONS, AND OPEN PROBLEMS16 with boundary condition (here Ω ⊆ R ) f ( x ) ≥ , (cid:90) Ω f ( x ) dx = 1 , Q ∗ (cid:90) Ω f ( x ) ζ ( x ) dx = λ. This equation defines a nonlinear eigenvalue problem for the eigenvalue λ = ˆ f Q ∗ .6. Discussions
Cancer is a group of diseases with abnormal cell growth. This paper provides a generalmathematical framework to describe the dynamics of heterogeneous stem cell regeneration,and introduce applications of the model to the study of cancer development. The modelhighlights the cell heterogeneity and plasticity, and provides a connections between hetero-geneity with cellular behavior, e.g. , proliferation, apoptosis, and differentiation/senescence.We also extend the model to include gene mutations that are important for cancer devel-opment. Nevertheless, many other factors may also play important roles in cancer devel-opment, and are not included in the proposed model, such as the cell-to-cell interactions,immune cells, microenvironment, spatial information, etc.
The current equation should beextended to included these factors for a more complete model.The proposed differential-integral equation model provides a general mathematical for-mulation for the process of stem cell regeneration. This equation integrates different scaleinteractions from epigenetic regulation to cell population dynamics. Mathematically, manybasic properties of the proposed differential-integral equation are not known and remainchallenge issues in future studies. Here we propose several mathematical problems thatare basic and important for our understanding of cancer development. We hope that theseequations can be the guideline for related studies in the future.Finally, the propose model is only a mathematical framework for stem cell regenerationwith heterogeneity and plasticity, detail implementation of the model should be subjectedto specific biological problems.
Acknowledgements
This research is supported by the National Natural Science Foundation of China (NSFC91730101and 11831015).
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