Analysis and Simulation of Division- and Label-Structured Population Models

Abstract

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.

Appendices

Appendix A: Proof of Analytical Solution of PDE (15)

To determine the solution of the PDE (15), the method of characteristics (Evans 1998) is employed, which is possible as (15) is linear. The characteristics of (15) are defined by the ODEs

(59)

with x(0)=x ₀, t(0)=0, and p _i (x ₀)=γ ⁱ p ₀(γ ⁱ x ₀). This system of ODEs has the solution

(60)

By substitution, we obtain

$$ \begin{aligned} p(x|i,t) &= \gamma^i e^{\int_0^t k(\tau) \,d\tau} p_{0}\bigl(\gamma^i e^{\int_0^t k(\tau) \,d \tau}x\bigr) \end{aligned} $$

(61)

as solution for (15).

Appendix B: Proof of Lemma 1: Solution of ODE System

In this section, we prove by mathematical induction that the ODE system

(62)

with initial conditions N(0|0)=N ₀ and ∀i≥1: N(i|0)=0, has for $\hat{\alpha},\check{\alpha} \geq 0$ and β>0 the solution:

(63)

Thereby, (62) is a generalization of (25).

It is trivial to verify that N(0|t) and N(1|t) are the solutions of (62) for i=0 and i=1, respectively. Hence, only the problem of proving that N(k+1|t) is the solution of (74) for i=k+1 given N(k|t) remains. To show this, note that

(64)

in which $\mathcal{N}(i|s)$ is the Laplace transform of N(i|t). Given this

(65)

Substitution of $\mathcal{N}(k|s)$ now yields,

(66)

which by applying the inverse Laplace transformation concludes the mathematical induction and proves Lemma 1.

Remark 6

Note that for $\check{\alpha} = \hat{\alpha} = \alpha$, (63) simplifies to (25). While for $\hat{\alpha} = \alpha_{\sup}$, $\check{\alpha} = \alpha_{\inf}$, β=β _inf, N(i|t)=B(i|t), and N ₀=B ₀, we obtain the bounding system (74) and its solution.

Appendix C: Proof of Lemma 2: Solution of ODE System

In this section, we prove that if

• ∀i: α _i (t)=α _i and β _i (t)=β _i and

• ∀i,j∈ℕ₀,i≠j: α _i +β _i ≠α _j +β _j

the solution of (14) is (26).

It is not difficult to verify that N(0|t) and N(1|t) are the solutions of (14) for i=0 and i=1, respectively. Hence, only the problem of proving that N(k+1|t) is the solution of (74) for i=k+1 given N(k|t) remains. To show this, note that for

(67)

in which $\mathcal{N}(i|s)$ is the Laplace transform of N(i|t). The proof of this relation is provided in Appendix D.

Given (67) it follows that

(68)

Substitution of $\mathcal{N}(k|s)$ now yields

(69)

which by applying the inverse Laplace transformation concludes the mathematical induction and proves (63).

Appendix D: Derivation of Laplace Transform $\mathcal{N}(i|s)$

To derive $\mathcal{N}(i|s)$ defined in (67), we study the partial fraction of

(70)

As under the prerequisite ∀i,j∈ℕ₀ with i≠j: α _i +β _i ≠α _j +β _j all poles are distinct, the partial fraction can be written as

(71)

To determine the coefficients c _k , we consider the equality constraint

(72)

As this equality constraint has to hold for all s, it must be satisfied for s=−(α _k +β _k ), yielding

$$ c_k = \Biggl( \prod_{\substack{j=1\\j\neq k}}^i \bigl((\alpha_j + \beta_j) - (\alpha_k + \beta_k)\bigr) \Biggr)^{-1}. $$

(73)

Given the values for c _k one can easily verify (67) by plugging in the c _k ’s into (71). Obviously, the proposed procedure can also be inverted, which concludes the derivation of (67).

Appendix E: Proof of Theorem 2: Convergence

To prove Theorem 2, the comparison theorem for series (Knopp 1964) is applied. Therefore, we define the bounding system

(74)

with initial conditions

and α _inf, α _sup, and β _inf as in Theorem 2. Due to the simple structure of (74), we can compute the analytical solution

(75)

whose derivation can be found in Appendix B.

The bounding system (74) is obtained from (14) by reducing the outflows out of and increasing the inflows into the individual subpopulations. Intuitively, as the initial conditions of (74) and (14) are identical and the right hand side of (74) is for every t∈[0,T] greater or equal than the right-hand side of (14), it follows that B _i is an upper bound for N _i ,

(76)

This can be proven rigorously by applying Müller’s theorem (Müller 1927), as shown in Kieffer and Walter (2011) for another system.

Given (75) and (76) one can prove the convergence of $\sum_{i\in\mathbb{N}_{0}} n(x,i|t)$. To take into account that a distributed process is considered (x≥0), we study the maximum over x and define

$$ B_i(t) := B(i|t) \gamma^i e^{kt} p^{\sup}_{0} = \frac{(2 \alpha_{\sup} \gamma)^i}{i!} t^i e^{-(\alpha_{\inf}+\beta_{\inf})t} e^{kt} n^{\sup}_{0} $$

(77)

with $p^{\sup}_{0} := \sup_{x \in \mathbb {R}_{+}} p_{0}(x)$ and $n^{\sup}_{0} := N_{0} p^{\sup}_{0}$. Thus, B _i (t) is a point-wise upper bound of n(x,i|t). For this definition of B _i (t) it holds that

• (i) ∀i,t,x≥0: 0≤N _i (t,x)≤B _i (t) ∀i, and

• (ii) the series

  $$ \begin{aligned} \sum_{i=0}^{\infty} B_i(t) &= \Biggl(\sum_{i=0}^{\infty} \frac{(2 \alpha_{\sup} \gamma t)^i}{i!} \Biggr) e^{-(\alpha_{\inf}+\beta_{\inf})t} e^{kt} n^{\sup}_{0} \end{aligned} $$

  (78)

  is convergent for every finite t.

The latter one holds true as the series is simply the Taylor expansion of the exponential $e^{2 \alpha_{\sup} \gamma t}$. Under conditions (i) and (ii), it follows from the comparison theorem for series (Knopp 1964) that the series $\sum_{i\in\mathbb{N}_{0}} N(i|t)$ is convergent in i for every t∈[0,T] and for every x≥0. This concludes the proof.

Appendix F: Proof of Theorem 3: Truncation Error

To prove Theorem 3, note that

(79)

in which the individual lines follow from the approximation methods (27), the fact that all quantities are positive, and the definition of the normalized label intensity (15) which has unity integral for all times T≥0. The remaining term in the following is successively upper bounded, for which we employ the bounding system (74). As shown in Appendix E, it holds that N(i|t)≤B(i|t) which yields

$$ \begin{aligned} \sum_{i = S}^\infty N(i|T) \leq \sum_{i = S}^\infty B(i|T) = \sum _{i = S}^\infty \frac{(2 \alpha_{\sup}T)^i}{i!} e^{-(\alpha_{\inf}+\beta_{\inf})T} N_{0}. \end{aligned} $$

(80)

By completion of the sum, this can be written as

$$ \begin{aligned} \sum_{i = S}^\infty N(i|T) & \leq \Biggl( e^{2 \alpha_{\sup} T} - \sum_{i = 0}^{S-1} \frac{(2 \alpha_{\sup}T)^i}{i!} \Biggr) e^{-(\alpha_{\inf}+\beta_{\inf})T} N_{0}. \end{aligned} $$

(81)

Thus, by exploiting that ∥n(x|0)∥₁=N ₀, one obtains (29), which concludes the proof.

Appendix G: Proof that the Solution of LSP Can Be Constructed from DLSP

To prove that the DLSP provides the solution to the LSP, n ^LSP(x|t)=n(x|t), we show that $n(x|t) = \sum_{i \in \mathbb{N}_{0}} N(i|t) p(x|i,t)$ solves (35). Therefore, n(x|t) is inserted in the left-hand side (∗) of (35), yielding

In here, dN(i|t)/dt is substituted with (14), resulting in

This is equivalent to the result if n(x|t) is inserted in the right-hand side (∗) of (35). Hence, $n(x|t) = \sum_{i \in \mathbb{N}_{0}} N(i|t) p(x|i,t)$ fulfills (35), which concludes the proof.

Appendix H: Proof that the PDE (15) Conserves Log-normal Distributions

To prove that the PDE (15) conserves log-normal distributions, we use its analytical solution (24) and consider $p_{0}(x) = \log\mathcal{N}(x|\mu_{0},\sigma_{0}^{2})$. This yields the solution

(82)

Employing the definition of the log-normal distribution, this equation becomes

(83)

(84)

for x>0, which can be restated as

(85)

in which $\mu_{i}(t) = - i \log \gamma - \int_{0}^{t} k(\tau)\, d \tau + \mu_{0}$. As this equation also holds for x≤0, it follows that the log-normal distribution is conserved and merely the parameter μ is time-dependent. Employing the superposition principle, this statement can be directly extended for sums of log-normal distributions, which concludes the proof.