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Mathematical models for CFSE labelled lymphocyte dynamics: asymmetry and time-lag in division

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Abstract

Since their invention in 1994, fluorescent dyes such as carboxyfluorescein diacetate succinimidyl ester (CFSE) are used for cell proliferation analysis in flow cytometry. Importantly, the interpretation of such assays relies on the assumption that the label is divided equally between the daughter cells upon cell division. However, recent experimental studies indicate that division of cells is not perfectly symmetric and there is unequal distribution of protein between sister cell pairs. The uneven partition of protein or mass to daughter cells can lead to an overlap in the generations of CFSE-labelled cells with straightforward consequences for the resolution of individual generations. Numerous mathematical models developed so far for the analysis of CFSE proliferation assay incorporate the premise that the CFSE fluorescence intensity is halved in the two daughter cells. Here, we propose a novel modelling approach for the analysis of the CFSE cell proliferation assays which are characterized by poorly resolved peaks of cell generations in flow cytometric histograms. We formulate a mathematical model in the form of a system of delay hyperbolic partial differential equations which provides a good agreement with the CFSE histograms time-series data and allows an analytical treatment. The model is a further generalization of the recently proposed class of division- and label-structured models as it considers an asymmetric cell division. In addition, the basic structure of the cell cycle, i.e. the resting and cycling cell compartments, is taken into account. The model is used to estimate fundamental parameters such as activation rate, duration of the cell cycle, apoptosis rate, CFSE decay rate and asymmetry factor in cell division of monoclonal T cells during cognate interaction with dendritic cells.

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Acknowledgments

The authors acknowledge the support of this work provided by the Swiss National Science Foundation, the Von-Tobel Foundation (Zurich), the Russian Foundation of Basic Research (11-01-00117a), the Programme of the Russian Academy of Sciences (Basic research for Medicine) and by the Swedish Institute, Visby Program.

Author information

Authors and Affiliations

  1. Institute of Mathematical Problems in Biology, RAS, Pushchino, Russia

    Tatyana Luzyanina

  2. Institute of Immunobiology, Kantonal Hospital St. Gallen, St. Gallen, Switzerland

    Jovana Cupovic & Burkhard Ludewig

  3. Institute of Numerical Mathematics, RAS, Moscow, Russia

    Gennady Bocharov

Authors
  1. Tatyana Luzyanina
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  2. Jovana Cupovic
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  3. Burkhard Ludewig
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  4. Gennady Bocharov
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Corresponding author

Correspondence to Tatyana Luzyanina.

Appendix

Appendix

In this appendix, we give the proof of Lemma 1 and Theorem 1.

Proof of Lemma 1

We use a sequential time integration of the chain of linear non-autonomous ordinary differential equations (which the delay equations are reduced to) to compute solutions \(N_0^r(t),~N_1^r(t)\) and \(N_i^c(t),~i=0,1,\ldots ,i_r-1,\) of the model (3). The method of mathematical induction is applied to compute solutions \(N_i^r(t),~i=2,\ldots ,i_r\).

To simplify formulas, let \(c_i: = \alpha _i+\beta _i,~c_i \ne c_k\) for \(i \ne k,~i=0,1,\ldots ,i_r\), \(T_i:= \sum _{j=0}^i \tau _j,~i=0,1,\ldots ,i_r-1\), \(F_i:= 2^i N^0 \prod _{j=0}^{i-1} \alpha _j\) and \(G_{i,j} := \prod _{k=0, k \ne j}^i(c_k-c_j)^{-1},~i=1,\ldots ,i_r\).

The first equation of model (3) is a linear ODE. Time integration of this equation gives

$$\begin{aligned} N_0^r(t)=N^0\mathrm{e}^{-c_0t}, \quad t\ge 0, \end{aligned}$$
(34)

where \(N^0=N_0^r(0)\) is the given initial condition. Substitution of (34) into the equation for \(N_1^r(t)\) gives a linear non-autonomous ODE,

$$\begin{aligned} \frac{\mathrm{d}N_1^r(t)}{\mathrm{d}t} = -c_1N_1^r(t)+2\alpha _0N_0^r(t-T_0)= -c_1N_1^r(t)+2\alpha _0 N^0\mathrm{e}^{-c_0(t-T_0)}. \end{aligned}$$
(35)

Time integration of this equation gives

$$\begin{aligned} N_1^r(t)&= 0, \quad t \in [0,T_0),\nonumber \\ N_1^r(t)&= 2\alpha _0 N^0(\mathrm{e}^{c_0(T_0-t)}(c_1-c_0)^{-1} + \mathrm{e}^{c_1(T_0-t)}(c_0-c_1)^{-1})\nonumber \\&= F_1 \sum _{j=0}^1 \mathrm{e}^{c_j (T_0 - t)} G_{1,j}, \quad t \ge T_0, \end{aligned}$$
(36)

where the first equality is due to the initial condition \(N_0^r(t-T_0)=0\) for \(t \in [0,T_0)\) which reduces non-autonomous ODE (35) to an autonomous ODE with zero initial condition \(N_1^r(0)=0\).

Now we assume that \(N_i^r(t)\), defined by

$$\begin{aligned} N_i^r(t) = 0, \quad t \in [0,T_{i-1}), \quad N_i^r(t) = F_i\sum _{j=0}^i \mathrm{e}^{c_j (T_{i-1} - t)} G_{i,j}, \quad t \ge T_{i-1}, \end{aligned}$$
(37)

solves the equation for \(N_i^r(t)\) in (3). We have to prove that

$$\begin{aligned} N_{i+1}^r(t) = 0, \quad t \in [0,T_{i}), \quad N_{i+1}^r(t) = F_{i+1} \sum _{j=0}^{i+1} \mathrm{e}^{c_j (T_{i} - t)} G_{i+1,j}, \quad t \ge T_{i}\qquad \end{aligned}$$
(38)

solves the equation for \(N_{i+1}^r(t)\) in (3). For this, we substitute (37) in the equation for \(N_{i+1}^r(t)\) in (3) and obtain

$$\begin{aligned} \frac{\mathrm{d}N_{i+1}^r(t)}{\mathrm{d}t} = -c_{i+1} N_{i+1}^r(t) + F_{i+1} \sum _{j=0}^{i} \mathrm{e}^{c_j (T_{i} - t)} G_{i,j}, \quad t \ge T_i, \end{aligned}$$
(39)

and \(N_{i+1}^r(t) = 0,~t \in [0,T_{i})\). Here we used that \(F_{i+1}=2 \alpha _{i} F_i\). Equation (39) is a linear ODE. Time integration of this equation (a number of standard steps) shows that \(N_{i+1}^r(t)\) defined by (38) is its solution.

The solutions \(N_i^c(t),~i=0,1,\ldots ,i_r-1,\) are obtained by the time integration of the corresponding equations. Namely, \(N_0^c(t)\) is computed as

$$\begin{aligned} N_0^c(t)= \alpha _0 \left( \int _0^t N_0^r(s)~\mathrm{d}s -\int _{0}^t N_0^r(s-T_0)~\mathrm{d}s\right) . \end{aligned}$$
(40)

Since \(N_0^r(t-T_0) = 0\) for \(t \in [0,T_0)\), the second integral in (40) is equal to zero for \(t \in [0,T_0)\) and we obtain

$$\begin{aligned} N_0^c(t)= \alpha _0 N^0 \int _0^t \mathrm{e}^{-c_0s}~\mathrm{d}s= \alpha _0 N^0 (1-\mathrm{e}^{-c_0 t})c_0^{-1}, \quad t \in [0,T_0), \end{aligned}$$
(41)

and

$$\begin{aligned} N_0^c(t)&= \alpha _0 N^0 \left( \int _0^t \mathrm{e}^{-c_0s}~\mathrm{d}s -\int _{T_0}^t\mathrm{e}^{-c_0(s-T_0)}~\mathrm{d}s\right) \nonumber \\&= \alpha _0 N^0 (\mathrm{e}^{c_0 (T_{0} - t)}-\mathrm{e}^{-c_0 t})c_0^{-1}, \quad t \ge T_0. \end{aligned}$$
(42)

Similarly, we integrate the equation for \(N_i^c(t),~i\ge 1\), using (37) and that \(N_i^r(t-\tau _i) = 0\) for \(t \in [0,T_i)\),

$$\begin{aligned} N_i^c(t)=\alpha _i \int _{T_{i-1}}^t N_i^r(s)~\mathrm{d}s = \alpha _i F_i \sum _{j=0}^i (1 - \mathrm{e}^{c_j(T_{i-1}-t)})c_j^{-1}G_{i,j}, \quad t \in [T_{i-1},T_i),\nonumber \\ \end{aligned}$$
(43)

and

$$\begin{aligned} N_i^c(t)&= \alpha _i \left( \int _{T_{i-1}}^t N_i^r(s)~\mathrm{d}s -\int _{T_i}^t N_i^r(s-\tau _i)~\mathrm{d}s\right) \nonumber \\&= \alpha _i F_i\left( \sum _{j=0}^i (1-\mathrm{e}^{c_j(T_{i-1}-t)}) c_j^{-1}G_{i,j}-\sum _{j=0}^i(1-\mathrm{e}^{c_j(T_{i}-t)})c_j^{-1} G_{i,j}\right) \nonumber \\&= \alpha _i F_i \sum _{j=0}^i (\mathrm{e}^{c_j(T_{i}-t)} - \mathrm{e}^{c_j(T_{i-1}-t)})c_j^{-1}G_{i,j},\quad t\ge T_i. \end{aligned}$$
(44)

The proof is complete. \(\square \)

Proof of Theorem 1

Although the idea of the proof is the same as the one for Theorem 1 in Schittler et al. (2011), Hasenauer et al. (2012a); Hasenauer et al. (2012b), we present the proof in detail since model (5) differs from the one in Schittler et al. (2011), Hasenauer et al. (2012a); Hasenauer et al. (2012b). The plan of the proof is the following: (a) we substitute the functions (13),

$$\begin{aligned}&n_i^r(t,x) = N_i^r(t) p_i(t,x),\quad i=0,1,\ldots ,i_r,\\&n_i^c(t,x) = N_i^c(t) p_i(t,x),\quad i=0,1,\ldots ,i_r-1, \end{aligned}$$

into model (5); (b) we use the Eqs. (3) and (14) to obtain, from the result of (a), a certain equality for \(p_i(t,x)\); (c) we solve the hyperbolic PDE (14) for \(p_i(t,x)\) to show that the equality for \(p_i(t,x)\) obtained in (b) is correct. This completes the proof. Below we present the proof for \(i \ge 1\), the proof for \(i=0\) is analogous.

(a) Substitution of the above expressions for \(n_i^r(t,x)\) and \(n_i^c(t,x)\) into model (5) gives

$$\begin{aligned}&\frac{\mathrm{d}N_i^r(t)}{\mathrm{d}t}p_i(t,x)+N_i^r(t)\left( \frac{\partial p_i(t,x)}{\partial t}-k \frac{\partial (x p_i(t,x))}{\partial x} \right) \nonumber \\&\quad =-(\alpha _i+\beta _i)N_i^r(t)p_i(t,x)+\alpha _{i-1}\mathrm{e}^{k \tau _{i-1}} \nonumber \\&\qquad \times \left( \frac{1}{m_1}N_{i-1}^r(t-\tau _{i-1})p_{i-1}\left( t-\tau _{i-1},\frac{e^{k \tau _{i-1}}x}{m_1}\right) \right. \nonumber \\&\qquad \left. +\frac{1}{m_2}N_{i-1}^r(t-\tau _{i-1})p_{i-1} \left( t-\tau _{i-1},\frac{e^{k \tau _{i-1}}x}{m_2}\right) \right) , \quad i=1,\ldots ,i_r, \end{aligned}$$
(45)

and

$$\begin{aligned}&\frac{\mathrm{d}N_i^c(t)}{\mathrm{d}t}p_i(t,x)+N_i^c(t)\left( \frac{\partial p_i(t,x)}{\partial t}-k \frac{\partial (x p_i(t,x))}{\partial x}\right) \nonumber \\&\quad =-\alpha _i (N_i^r(t)p_i(t,x)-\mathrm{e}^{k \tau _{i}}N_i^r(t-\tau _i)p_i(t-\tau _i,\mathrm{e}^{k \tau _i}x)),\quad i=1,\ldots ,i_r-1.\nonumber \\ \end{aligned}$$
(46)

(b) Putting the term in brackets on the left hand side of equations (45)–(46) to zero due to (14) and using the Eq. (3) for \(\mathrm{d}N_i^r(t)/\mathrm{d}t\) and \(\mathrm{d}N_i^c(t)/\mathrm{d}t\), the expressions (45)–(46) are reduced to

$$\begin{aligned}&p_i(t,x)\nonumber \\&\quad = \frac{\mathrm{e}^{k \tau _{i-1}}}{2} \left( \frac{1}{m_1} p_{i-1}\left( t-\tau _{i-1},\frac{\mathrm{e}^{k\tau _{i-1}}x}{m_1}\right) +\frac{1}{m_2} p_{i-1}\left( t-\tau _{i-1},\frac{\mathrm{e}^{k \tau _{i-1}} x}{m_2}\right) \right) ,\nonumber \\ \end{aligned}$$
(47)

respectively,

$$\begin{aligned} p_i(t,x) = \mathrm{e}^{k \tau _i} p_i(t-\tau _i,\mathrm{e}^{k \tau _i}x). \end{aligned}$$
(48)

(c) The method of characteristics applied to hyperbolic PDE (14) gives

$$\begin{aligned} p_i(t,x)=\mathrm{e}^{kt}p_i(0,\mathrm{e}^{kt}x), \quad i=0,1,\ldots , i_r. \end{aligned}$$
(49)

Using the initial condition for \(p_i(0,x)\) from (15), the solution \(p_i(t,x)\) of PDE (14) is

$$\begin{aligned} p_i(t,x)&= \mathrm{e}^{kt}p_i(0,\mathrm{e}^{kt}x)\nonumber \\&= \frac{\mathrm{e}^{k t}}{2}\left( \frac{1}{m_1} p_{i-1}\left( 0,\frac{\mathrm{e}^{k t}x}{m_1}\right) +\frac{1}{m_2} p_{i-1}\left( 0,\frac{\mathrm{e}^{kt}x}{m_2}\right) \right) ,\quad i=1,\ldots ,i_r.\nonumber \\ \end{aligned}$$
(50)

As follows from (49),

$$\begin{aligned} p_{i-1}\left( t-\tau _{i-1},\frac{\mathrm{e}^{k\tau _{i-1}}x}{m_1}\right) = \mathrm{e}^{k(t-\tau _{i-1})} p_{i-1}\left( 0,\frac{\mathrm{e}^{kt}x}{m_1}\right) \end{aligned}$$
(51)

and

$$\begin{aligned} p_{i-1}\left( t-\tau _{i-1},\frac{\mathrm{e}^{k\tau _{i-1}}x}{m_2}\right) = \mathrm{e}^{k(t-\tau _{i-1})} p_{i-1}\left( 0,\frac{\mathrm{e}^{kt}x}{m_2}\right) . \end{aligned}$$
(52)

If we multiply (51) and (52) by \(\mathrm{e}^{k\tau _{i-1}}/2m_1\) and \(\mathrm{e}^{k\tau _{i-1}}/2m_2\), respectively, and sum the two equalities, we obtain,

$$\begin{aligned}&\frac{\mathrm{e}^{k \tau _{i-1}}}{2}\left( \frac{1}{m_1} p_{i-1}\left( t-\tau _{i-1},\frac{\mathrm{e}^{k \tau _{i-1}}x}{m_1}\right) + \frac{1}{m_2} p_{i-1}\left( t-\tau _{i-1},\frac{\mathrm{e}^{k \tau _{i-1}}x}{m_2}\right) \right) \nonumber \\&\quad =\frac{\mathrm{e}^{kt}}{2}\left( \frac{1}{m_1} p_{i-1}\left( 0,\frac{\mathrm{e}^{kt}x}{m_1}\right) + \frac{1}{m_2}p_{i-1}\left( 0,\frac{\mathrm{e}^{kt}x}{m_2}\right) \right) =p_i(t,x), \end{aligned}$$
(53)

where the last equality is due to (50). Hence, the equality (47) is correct and \(n_i^r(t,x) = N_i^r(t) p_i(t,x),~i=1,\ldots ,i_r\), is the solution of model (5).

The equality (48) is correct due to (49). Namely,

$$\begin{aligned} \mathrm{e}^{k \tau _i} p_i(t-\tau _i,\mathrm{e}^{k \tau _i}x) = \mathrm{e}^{k \tau _i}( \mathrm{e}^{k(t-\tau _i)} p_i(0,\mathrm{e}^{k\tau _i}\mathrm{e}^{k(t-\tau _i)}x))= \mathrm{e}^{kt}p_i(0,\mathrm{e}^{kt}x)= p_i(t,x). \end{aligned}$$

Hence, \(n_i^c(t,x)=N_i^c(t) p_i(t,x),~i=1,\ldots ,i_r-1,\) is the solution of model (5). The proof is complete. \(\square \)

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Luzyanina, T., Cupovic, J., Ludewig, B. et al. Mathematical models for CFSE labelled lymphocyte dynamics: asymmetry and time-lag in division. J. Math. Biol. 69, 1547–1583 (2014). https://doi.org/10.1007/s00285-013-0741-z

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