Multi-type Galton–Watson Processes with Affinity-Dependent Selection Applied to Antibody Affinity Maturation

Abstract

We analyze the interactions between division, mutation and selection in a simplified evolutionary model, assuming that the population observed can be classified into fitness levels. The construction of our mathematical framework is motivated by the modeling of antibody affinity maturation of B-cells in germinal centers during an immune response. This is a key process in adaptive immunity leading to the production of high-affinity antibodies against a presented antigen. Our aim is to understand how the different biological parameters affect the system’s functionality. We identify the existence of an optimal value of the selection rate, able to maximize the number of selected B-cells for a given generation.

Appendices

Appendix

Appendix A: Few Reminders of Classical Results on GW Processes

We recall here some classical results about GW processes we employed to derive Proposition 1 (Sect. 3.1). For further details, the reader can refer to Harris (1963).

Definition 14

Let X be an integer valued rv, $p_k:={\mathbb {P}}(X=k)$ for all $k\ge 0$. Its probability generating function (pgf) is given by:

$$\begin{aligned} F_X(s)=\sum _{k=0}^{+\infty } p_k s^k \end{aligned}$$

$F_X$ is a convex monotonically increasing function over [0, 1], and $F_X(1)=1$. If $p_0\ne 0$ and $p_0+p_1<1$ then F is a strictly increasing function.

Definition 15

Given F, the pgf of a rv X, the iterates of F are given by:

$$\begin{aligned} F_0(s)= & {} s \\ F_1(s)= & {} F(s) \\ F_t(s)= & {} F(F_{t-1}(s))\text { for }t\ge 2 \end{aligned}$$

Proposition 11

• (i) If ${\mathbb {E}}(X)$ exists (respectively ${\mathbb {V}}(X)$), then ${\mathbb {E}}(X)=F'_X(1)$ (respectively ${\mathbb {V}}(X)=F''_X(1)-\left( {\mathbb {E}}(X)\right) ^2+{\mathbb {E}}(X)$).

• (ii) If X and Y are two integer valued independent rvs, then $X+Y$ is still an integer valued rv and its pgf is given by $F_{X+Y}=F_XF_Y$.

Definition 16

We denote by $\eta$ the extinction probability of the process $(Z_t)_{t\in {\mathbb {N}}}$:

$$\begin{aligned} \eta :=\lim _{t\rightarrow \infty }F_t(0) \end{aligned}$$

Theorem 2

• (i) The pgf of $Z_t^{(z_0)}$, $t\;\in \;{\mathbb {N}}$, which represents the population size of the tth-generation starting from $z_0\ge 1$ seed cells, is $F_t^{(z_0)}=(F_t)^{z_0}$, $F_t$ being the tth-iterate of F (Eq. (2)).

• (ii) The expected size of the GC at time t and starting from $z_0$ B-cells is given by:

  $$\begin{aligned} {\mathbb {E}}(Z_t^{(z_0)})=z_0\left( {\mathbb {E}}(Z_t)\right) =z_0\left( {\mathbb {E}}(Z_1)\right) ^t~, \end{aligned}$$

  (13)

• (iii) $\eta$ is the smallest fixed point of the generating function F, i.e., $\eta$ is the smallest s s.t. $F(s)=s$.

• (iv) If ${\mathbb {E}}(Z_1)=:m$ is finite, then:

  • if $m\le 1$ then F has only 1 as fixed point and consequently $\eta =1$;

  • if $m>1$ then F as exactly a fixed point on [0, 1[ and then $\eta <1$.

• (v) Denoted by $\eta _{z_0}$ the probability of extinction of $(Z_t^{(z_0)})$, one has:

  $$\begin{aligned} \eta _{z_0}=\eta ^{z_0} \end{aligned}$$

  where $\eta$ is given by (iii).

Proposition 1 of Sect. 3.1 follows by applying Theorem 2 and Eq. (1).

Appendix B: Proof of Proposition 2

For all $j\in \{0,\dots ,N+2\}$ the generating function of $Z_j$ gives the number of offspring of each type that a type j particle can produce. It is defined as follows:

$$\begin{aligned} f^{(j)}(s_0,\ldots ,s_{N+2})=\sum _{k_0,\ldots ,k_{N+2}\ge 0}p^{(j)}(k_0,\ldots ,k_{N+2})s_0^{k_0}\ldots s_{N+2}^{k_{N+2}}, \end{aligned}$$

(14)

$$\begin{aligned} 0\le s_{\alpha }\le 1\text { for all }\alpha \in \{0,\dots ,N+2\} \end{aligned}$$

where $p^{(j)}(k_0,\dots ,k_{N+2})$ is the probability that a type j cell produces $k_0$ cells of type 0, $k_1$ of type 1, $\dots$, $k_{N+2}$ of type $N+2$ for the next generation.

We denote:

• ${\mathbf {p}}({\mathbf {k}})=(p^{(0)}({\mathbf {k}}),\dots ,p^{(N+2)}({\mathbf {k}}))$, for ${\mathbf {k}}=(k_0,\dots ,k_{N+2}) \,\in \,{\mathbb {Z}}_+^{N+3}$

• ${\mathbf {f}}({\mathbf {s}})=(f^{(1)}({\mathbf {s}}),\dots ,f^{(N+1)}({\mathbf {s}}))$, for ${\mathbf {s}}=(s_0,\dots ,s_{N+2})\,\in \,{\mathcal {C}}^{N+3}:=[0,1]^{N+3}$

Then the probability generating function of ${\mathbf {Z}}_1$ is given by:

$$\begin{aligned} {\mathbf {f}}({\mathbf {s}})=\displaystyle \sum _{{\mathbf {k}} \in {\mathbb {Z}}_+^{N+3}}{\mathbf {p}}({\mathbf {k}}){\mathbf {s}}^{{\mathbf {k}}} {,{\mathbf {s}}\,\in \,{\mathcal {C}}^{N+3}} \end{aligned}$$

(15)

Again, the generating function of ${\mathbf {Z}}_t$, ${\mathbf {f}}_t({\mathbf {s}})$, is obtained as the tth-iterate of ${\mathbf {f}}$, and it holds true that:

$$\begin{aligned} {\mathbf {f}}_{t+r}({\mathbf {s}})={\mathbf {f}}_t[{\mathbf {f}}_r({\mathbf {s}})]{, {\mathbf {s}}\,\in \,{\mathcal {C}}^{N+3}.} \end{aligned}$$

Let $m_{ij}:={\mathbb {E}}[Z_{1,j}^{(i)}]$ the expected number of offspring of type j of a cell of type i in one generation. We collect all $m_{ij}$ in a matrix, ${\mathcal {M}}=(m_{ij})_{0\le i,j\le N+2}$. We have Athreya and Ney (2012):

$$\begin{aligned} m_{ij}=\frac{\partial f^{(i)}}{\partial s_j}({\varvec{1}}) \end{aligned}$$

and:

$$\begin{aligned} {\mathbb {E}}[Z_{t,j}^{(i)}]=\frac{\partial f_t^{(i)}}{\partial s_j}({\varvec{1}}) \end{aligned}$$

(16)

Finally:

$$\begin{aligned} {\mathbb {E}}[{\mathbf {Z}}_t^{({\mathbf {i}})}]={\mathbf {i}}{\mathcal {M}}^t \end{aligned}$$

(17)

One can explicitly derive the elements of matrix ${\mathcal {M}}$ for the process described in Definition 13.

Proposition

${\mathcal {M}}$ is a $(N+3)\times (N+3)$ matrix defined as a block matrix:

$$\begin{aligned} {\mathcal {M}}=\left( \begin{array}{cc} {\mathcal {M}}_1 &{}\quad {\mathcal {M}}_2 \\ {\varvec{0}}_{2\times (N+1)} &{}\quad {\mathcal {I}}_2 \end{array}\right) \end{aligned}$$

where

• ${\varvec{0}}_{2\times (N+1)}$ is a $2\times (N+1)$ matrix with all entries 0;

• ${\mathcal {I}}_n$ is the identity matrix of size n;

• ${\mathcal {M}}_1=2(1-r_d)r_\mathrm{div}(1-r_s){\mathcal {Q}}_N+(1-r_d)(1-r_\mathrm{div})(1-r_s){\mathcal {I}}_{N+1}$

• ${\mathcal {M}}_2=(m_{2,ij})$ is a $(N+1)\times 2$ matrix where for all $i\in \{0,\dots ,N\}$:

  • if $i\le {\overline{a}}_s$:

    $m_{2,i1}=(1-r_d)(1-r_\mathrm{div})r_s+2(1-r_d)r_\mathrm{div}r_s\sum _{j=0}^{{\overline{a}}_s}q_{ij}$,

    $m_{2,i2}=r_d+2(1-r_d)r_\mathrm{div}r_s\sum _{j={\overline{a}}_s+1}^{N}q_{ij}$

  • if $i>{\overline{a}}_s$:

    $m_{2,i1}=2(1-r_d)r_\mathrm{div}r_s\sum _{j=0}^{{\overline{a}}_s}q_{ij}$,

    $m_{2,i2}=r_d+(1-r_d)(1-r_\mathrm{div})r_s+2(1-r_d)r_\mathrm{div}r_s \sum _{j={\overline{a}}_s+1}^{N}q_{ij}$

Proof

One has to compute all $f^{(i)}({\mathbf {s}})$ for $i=0,\dots , N+2$, which depend on $r_d$, $r_\mathrm{div}$, $r_s$, ${\overline{a}}_s$ and the elements of ${\mathcal {Q}}_N$. First, the elements of the $(N+2)$th and $(N+3)$th-lines are obviously determined: all selected (resp. dead) cells remain selected (resp. dead) for next generations, as they cannot give rise to any other cell type offspring (we do not take into account here any type of recycling mechanism). Let $i\in \{0,\dots ,N\}$ be a fixed index: we evaluate $m_{ij}$ for all $j\in \{0,\dots ,N+2\}$. The first step is to determine the value of $p^{(i)}({\mathbf {k}})$ for ${\mathbf {k}}=(k_0,\dots ,k_{N+2}) \,\in \,{\mathbb {Z}}_+^{N+3}$. There exists only a few cases in which $p^{(i)}({\mathbf {k}})\ne 0$, which can be explicitly evaluated:

• $p^{(i)}(0,\dots ,0,1)={\left\{ \begin{array}{ll} r_d &{}\quad \text {if}\; i \le {\overline{a}}_s \\ r_d+(1-r_d)(1-r_\mathrm{div})r_s &{}\quad \text {otherwise} \end{array}\right. }$

• $p^{(i)}(0,\dots ,0,1,0)={\left\{ \begin{array}{ll} (1-r_d)(1-r_\mathrm{div})r_s &{}\quad \text {if}\; i \le {\overline{a}}_s \\ 0 &{}\quad \text {otherwise} \end{array}\right. }$

• $p^{(i)}(0,\dots ,0,\underset{{i}}{1},0,\dots ,0,0)=(1-r_d)(1-r_\mathrm{div})(1-r_s)$

• $p^{(i)}(0,\dots ,0,2)=(1-r_d)r_\mathrm{div}r_s^2\sum _{j_1={\overline{a}}_s+1}^N q_{ij_1}\sum _{j_2={\overline{a}}_s+1}^N q_{ij_2}$

• $p^{(i)}(0,\dots ,0,2,0)=(1-r_d)r_\mathrm{div}r_s^2\sum _{j_1=0}^{{\overline{a}}_s} q_{ij_1}\sum _{j_2=0}^{{\overline{a}}_s} q_{ij_2}$

• $p^{(i)}(0,\dots ,0,1,1)=2(1-r_d)r_\mathrm{div}r_s^2\sum _{j_1=0}^{{\overline{a}}_s} q_{ij_1}\sum _{j_2={\overline{a}}_s+1}^N q_{ij_2}$

• For all $j_1<j_2\in \{0,\dots ,N\}$:

  • $p^{(i)}(0,\dots ,0,\underset{{j_1}}{2},0,\dots ,0,0)=(1-r_d)r_\mathrm{div}(1-r_s)^2q_{ij_1}^2$

  • $p^{(i)}(0,\dots ,0,\underset{{j_1}}{1},0,\dots ,0,\underset{{j_2}}{1},0,\dots ,0,0)=2(1-r_d)r_\mathrm{div}(1-r_s)^2q_{ij_1}q_{ij_2}$

  • $p^{(i)}(0,\dots ,0,\underset{{j_1}}{1},0,\dots ,0,1)=2(1-r_d)r_\mathrm{div}r_s(1-r_s)q_{ij_1}\sum _{j_2={\overline{a}}_s+1}^N q_{ij_2}$

  • $p^{(i)}(0,\dots ,0,\underset{{j_1}}{1},0,\dots ,0,1,0)=2(1-r_d)r_\mathrm{div}r_s(1-r_s)q_{ij_1}\sum _{j_2=0}^{{\overline{a}}_s} q_{ij_2}$

• $p^{(i)}({\mathbf {k}})=0$ otherwise

We can therefore evaluate $f^{(i)}({\mathbf {s}})$, with ${\mathbf {s}}=(s_0,\dots ,s_{N+2})\in {\mathcal {C}}^{N+3}$.

For all $i\le {\overline{a}}_s$:

$$\begin{aligned} \begin{aligned} f^{(i)} ({\mathbf {s}})&= r_ds_{N+2}+(1-r_d)(1-r_\mathrm{div})r_ss_{N+1} + (1-r_d)(1-r_\mathrm{div})(1-r_s)s_i \\&\quad + (1-r_d)r_\mathrm{div}r_s^2\left( \sum _{j_1={\overline{a}}_s+1}^N q_{ij_1}\sum _{j_2={\overline{a}}_s+1}^N q_{ij_2}s_{N+2}^2\right. \\&\quad \left. +\sum _{j_1=0}^{{\overline{a}}_s} q_{ij_1}\sum _{j_2=0}^{{\overline{a}}_s} q_{ij_2}s_{N+1}^2+2\sum _{j_1=0}^{{\overline{a}}_s} q_{ij_1}\sum _{j_2={\overline{a}}_s+1}^N q_{ij_2}s_{N+1}s_{N+2}\right) \\&\quad + (1-r_d)r_\mathrm{div}(1-r_s)^2\left( \sum _{j_1=0}^{N}q_{ij_1}^2s_{j_1}^2+2\sum _{j_1=0}^{N}q_{ij_1}\sum _{j_2< j_1=0}^N q_{ij_2}s_{j_1}s_{j_2}\right) \\&\quad + 2(1-r_d)r_\mathrm{div}r_s(1-r_s)\sum _{j_1=0}^{N}q_{ij_1}\left( \displaystyle \sum _{j_2={\overline{a}}_s+1}^N q_{ij_2}s_{N+2}+\sum _{j_2=0}^{{\overline{a}}_s} q_{ij_2}s_{N+1}\right) s_{j_1} \end{aligned} \end{aligned}$$

(18)

If $i>{\overline{a}}_s$ then $f^{(i)}({\mathbf {s}})$ is the same except for the first line, which becomes:

$$\begin{aligned} (r_d+(1-r_d)(1-r_\mathrm{div})r_s)s_{N+2} + (1-r_d)(1-r_\mathrm{div})(1-r_s)s_i \end{aligned}$$

The values of each $m_{ij}$ are now obtained by evaluating all partial derivatives of $f^{(i)}({\mathbf {s}})$ in ${\varvec{1}}$, keeping in mind that for all $i\;\in \;\{0,\dots ,N\}$, $\sum _{j=0}^N q_{ij}=1$. $\square$

Appendix C: Deriving the Extinction Probability of the GC from the Multi-type GW Process (Sect. 3.2)

Let us recall some results about the extinction probability for multi-type GW processes (Athreya and Ney 2012).

Definition 17

Let $q^{(i)}$ be the probability of eventual extinction of the process, when it starts from a single type i cell. As above bold symbols denote vectors, i.e., ${\mathbf {q}} :=(q^{(0)},\dots ,q^{(N+2)})\ge 0$.

Definition 18

We say that $({\mathbf {Z}}_t)$ is singular if each particle has exactly one offspring, which implies that the branching process becomes a simple MC.

Definition 19

Matrix ${\mathcal {M}}$ is said to be strictly positive if it has nonnegative entries and there exists a t s.t. $\left( {\mathcal {M}}^t\right) _{ij}>0$ for all i, j. $({\mathbf {Z}}_t)$ is called positive regular iff ${\mathcal {M}}$ is strictly positive.

Notation 1

Let ${\mathbf {u}}$, ${\mathbf {v}}\;\in \;{\mathbb {R}}^n$. We say that ${\mathbf {u}}\le {\mathbf {v}}$ if $u_i\le v_i$ for all $i\;\in \,\{1,\dots ,n\}$. Moreover, we say that ${\mathbf {u}}< {\mathbf {v}}$ if ${\mathbf {u}}\le {\mathbf {v}}$ and ${\mathbf {u}}\ne {\mathbf {v}}$.

Theorem 3

Let $({\mathbf {Z}}_t)$ be non-singular and strictly positive. Let $\rho$ be the maximal eigenvalue of ${\mathcal {M}}$. The following three results hold:

• 1. If $\rho <1$ (subcritical case) or $\rho =1$ (critical case) then ${\mathbf {q}}={\varvec{1}}$. Otherwise, if $\rho >1$ (supercritical case), then ${\mathbf {q}}<{\varvec{1}}$.

• 2. $\displaystyle \lim _{t\rightarrow \infty }{\mathbf {f}}_t({\mathbf {s}})={\mathbf {q}}$, for all ${\mathbf {s}}\,\in \,{\mathcal {C}}^{N+3}$.

• 3. ${\mathbf {q}}$ is the only solution of ${\mathbf {f}}({\mathbf {s}})={\mathbf {s}}$ in ${\mathcal {C}}^{N+3}$.

The spectrum of matrix ${\mathcal {M}}$ defined in Definition 2 (and recalled in “Appendix B”) is obtained as follows:

Proposition 12

Let ${\mathcal {M}}$ be defined as a block matrix as in Proposition 2. Let $\lambda _{{\mathcal {M}},i}$ be its ith-eigenvalue. The spectrum of ${\mathcal {M}}$ is given by:

• For all $i\in \{0,\dots ,N\}$, $\lambda _{{\mathcal {M}},i}=(1-r_d)(1-r_s)(1+r_\mathrm{div}(2\lambda _{i}-1))$, where $\lambda _{i}$ is the ith-eigenvalue of matrix ${\mathcal {Q}}_N$.

• whereas $\lambda _{{\mathcal {M}},N+1}=1$ with multiplicity 2.

Proof

As ${\mathcal {M}}$ is a block matrix with the lower left block composed of zeros, then $Spec({\mathcal {M}})=Spec({\mathcal {M}}_1)\cup Spec({\mathcal {I}}_{2})$. The result follows. $\square$

Therefore we obtain the same condition as in Proposition 1 for the extinction probability in the GC:

Proposition 13

Let ${\mathbf {q}}$ be the extinction probability for the process $({\mathbf {Z}}_t)$ defined in Definition 13 and restricted to the first $N+1$ components (i.e., we refer only to matrix ${\mathcal {M}}_1$, which defines the expectations of GC B-cells). Therefore:

• if $r_s\ge 1-\displaystyle \frac{1}{(1-r_d)(1+r_\mathrm{div})}$, then ${\mathbf {q}}={\varvec{1}}$

• otherwise ${\mathbf {q}}<{\varvec{1}}$ is the smallest fixed point of ${\mathbf {f}}({\mathbf {s}})$ in ${\mathcal {C}}^{N+3}$.

Proof

${\mathcal {Q}}_N$ is a stochastic matrix, therefore its largest eigenvalue is 1. The corresponding eigenvalue of matrix ${\mathcal {M}}_1$ is: $\lambda _{{\mathcal {M}}_1,1}=(1-r_d)(1-r_s)(1+r_\mathrm{div})$. The proposition is proved by observing that $\lambda _{{\mathcal {M}}_1,1}\le 1\Leftrightarrow r_s\ge 1-\frac{1}{(1-r_d)(1+r_\mathrm{div})}$ and applying Theorem 3 (note that ${\mathcal {M}}_1$ is positive regular: this is not the case for matrix ${\mathcal {M}}$). $\square$

Appendix D: Expected Size of the GC Derived from the Multi-type GW Process (Sect. 3.2)

Proposition

Let ${\mathbf {i}}$ be the initial state, $z_0:=|{\mathbf {i}}|$ its 1-norm ($|{\mathbf {i}}|:=\sum _{j=0}^{N+2}{\mathbf {i}}_j$). The expected size of the GC at time t:

$$\begin{aligned} \sum _{k=0}^N({\mathbf {i}}{\mathcal {M}}^t)_k=|{\mathbf {i}}|\left( (1-r_d)(1+r_\mathrm{div})(1-r_s)\right) ^{t} \end{aligned}$$

Proof

For the sake of simplicity, let us suppose that the process starts from a single B-cell belonging to the affinity class $a_0=i$ with respect to the target trait. We do not need to specify the transition probability matrix used to define the mutational model allowed.

We recall the expression of ${\mathcal {M}}^t$ obtained by iteration:

$$\begin{aligned} {{\mathcal {M}}}^t=\left( \begin{array}{cc} {\mathcal {M}}_1^t &{}\quad \sum _{k=0}^{t-1}{\mathcal {M}}_1^k{\mathcal {M}}_2 \\ \\ {\varvec{0}}_{2\times (N+1)} &{}\quad {\mathcal {I}}_2 \end{array}\right) \end{aligned}$$

Therefore we can claim that $({\mathbf {i}}{\mathcal {M}}^t)_k$ corresponds to the kth-component of the ith-row of matrix ${\mathcal {M}}_1^t=(2(1-r_d)r_\mathrm{div}(1-r_s){\mathcal {Q}}_N+(1-r_d)(1-r_\mathrm{div})(1-r_s){\mathcal {I}}_{N+1})^t$, where ${\mathcal {Q}}_N$ is a stochastic matrix. Matrices ${\mathcal {A}}:=2(1-r_d)r_\mathrm{div}(1-r_s){\mathcal {Q}}_N$ and ${\mathcal {B}}:=(1-r_d)(1-r_\mathrm{div})(1-r_s){\mathcal {I}}_{N+1}$ clearly commute, therefore:

$$\begin{aligned} \left( {\mathcal {A}}+{\mathcal {B}}\right) ^t=\sum _{j=0}^tC_t^j{\mathcal {A}}^{t-j}{\mathcal {B}}^j \end{aligned}$$

(19)

For all j, $0\le j\le t$:

$$\begin{aligned} {\mathcal {A}}^{t-j}{\mathcal {B}}^j= & {} 2^{t-j}(1-r_d)^{t-j}r_\mathrm{div}^{t-j}(1-r_s)^{t-j}(1-r_d)^j(1-r_\mathrm{div})^j(1-r_s)^j{\mathcal {Q}}_N^{t-j} \\= & {} (1-r_d)^t(1-r_s)^t(2r_\mathrm{div})^{t-j}(1-r_\mathrm{div})^j{\mathcal {Q}}_N^{t-j} \end{aligned}$$

Hence

$$\begin{aligned} \left( {\mathcal {A}}+{\mathcal {B}}\right) ^t=(1-r_d)^t(1-r_s)^t\sum _{j=0}^tC_t^j(2r_\mathrm{div})^{t-j}(1-r_\mathrm{div})^j{\mathcal {Q}}_N^{t-j} \end{aligned}$$

And consequently

$$\begin{aligned} \sum _{k=0}^N({\mathbf {i}}{\mathcal {M}}^t)_k= & {} \sum _{k=0}^N\left( {\mathbf {i}}\left( {\mathcal {A}}+{\mathcal {B}}\right) ^t\right) _k\\= & {} (1-r_d)^t(1-r_s)^t\sum _{j=0}^tC_t^j(2r_\mathrm{div})^{t-j}(1-r_\mathrm{div})^j\sum _{k=0}^N\left( {\mathbf {i}}{\mathcal {Q}}_N^{t-j}\right) _k \end{aligned}$$

Since ${\mathcal {Q}}_N$ is a stochastic matrix, for all n, ${\mathcal {Q}}_N^n$ is still a stochastic matrix, i.e., the entries of each row of ${\mathcal {Q}}_N^n$ sum to 1. Therefore:

$$\begin{aligned} \sum _{k=0}^N({\mathbf {i}}{\mathcal {M}}^t)_k= & {} (1-r_d)^t(1-r_s)^t\sum _{j=0}^tC_t^j(2r_\mathrm{div})^{t-j}(1-r_\mathrm{div})^j \\= & {} (1-r_d)^t(1-r_s)^t(2r_\mathrm{div}+1-r_\mathrm{div})^t = (1-r_d)^t(1-r_s)^t(1+r_\mathrm{div})^t~, \end{aligned}$$

as stated by Eq. (3) for $z_0=1$. This result can be easily generalized to the case of $z_0\ge 1$ initial B-cells. $\square$

Appendix E: Proof of Proposition 5

Proposition

Let us suppose that at time $t=0$ there is a single B-cell entering the GC belonging to the ith-affinity class with respect to the target cell. Moreover, let us suppose that ${\mathcal {Q}}_N=R\varLambda _N L$. For all $t\ge 1$, the expected number of selected B-cells at time t, is:

$$\begin{aligned} {\mathbb {E}}(S_t)=r_s(1-r_s)^{t-1}(1-r_d)^t\sum _{\ell =0}^N(2\lambda _{\ell }r_\mathrm{div}+1-r_\mathrm{div})^t \sum _{k=0}^{{\overline{a}}_s}r_{i\ell }l_{\ell k}~, \end{aligned}$$

Proof

Let us suppose, for the sake of simplicity, that ${\mathcal {Q}}_N$ is diagonalizable:

$$\begin{aligned} {\mathcal {Q}}_N=R\varLambda _N L~, \end{aligned}$$

(20)

We can prove by iteration that:

$$\begin{aligned} {{\mathcal {M}}}^t=\left( \begin{array}{cc} {\mathcal {M}}_1^t &{}\quad \sum _{k=0}^{t-1}{\mathcal {M}}_1^k{\mathcal {M}}_2 \\ \\ {\varvec{0}}_{2\times (N+1)} &{}\quad {\mathcal {I}}_2 \end{array}\right) \end{aligned}$$

(21)

It follows from (20) and (21) that for all $t\ge 1$, ${\mathcal {M}}^t$ can be written as:

$$\begin{aligned} {\mathcal {M}}^t=\left( \begin{array}{cc} RD^tL &{}\quad \left( R\sum _{k=0}^{t-1}D^kL\right) {\mathcal {M}}_2 \\ \\ {\varvec{0}}_{2\times (N+1)} &{}\quad {\mathcal {I}}_2 \end{array}\right) ~, \end{aligned}$$

(22)

where $D=2(1-r_d)r_\mathrm{div}(1-r_s)\varLambda _N+(1-r_d)(1-r_\mathrm{div})(1-r_s){\mathcal {I}}_{N+1}$ is a diagonal matrix. We obtain its expression thanks to Proposition 2.

Moreover, by Proposition 3 and Eq. (20) we have:

$$\begin{aligned} \widetilde{{\mathcal {M}}}=\left( \begin{array}{cc} R\widetilde{D}L &{}\quad \widetilde{{\mathcal {M}}}_2 \\ {\varvec{0}}_{2\times (N+1)} &{}\quad {\mathcal {I}}_2 \end{array}\right) ~, \end{aligned}$$

(23)

where $\widetilde{D}=2(1-r_d)r_\mathrm{div}\varLambda _N+(1-r_d)(1-r_\mathrm{div}){\mathcal {I}}_{N+1}$ is a diagonal matrix.

Proposition 4 claims:

$$\begin{aligned} {\mathbb {E}}(S_t)=r_s\displaystyle \sum _{k=0}^{{\overline{a}}_s}\left( {\mathbf {i}}{\mathcal {M}}^{t-1}\widetilde{{\mathcal {M}}}\right) _k \end{aligned}$$

From Eqs. (22) and (23):

$$\begin{aligned} {\mathcal {M}}^{t-1}\widetilde{{\mathcal {M}}}=\left( \begin{array}{cc} RD^{t-1}\widetilde{D}L &{}\quad RD^{t-1}L\widetilde{{\mathcal {M}}}_2+\left( R\sum _{k=0}^{t-2}D^kL\right) {\mathcal {M}}_2 \\ {\varvec{0}}_{2\times (N+1)} &{}\quad {\mathcal {I}}_2 \end{array}\right) \end{aligned}$$

Since, by hypothesis, $\mathbf {i}=(0,\dots ,0,1,0,\dots ,0,0)$, with the only 1 being at position i, $0\le i\le N$, then $\left( {\mathbf {i}}{\mathcal {M}}^{t-1}\widetilde{{\mathcal {M}}}\right)$ denotes the ith-row of matrix ${\mathcal {M}}^{t-1}\widetilde{{\mathcal {M}}}$. Therefore, we are interested in the sum between 0 and ${\overline{a}}_s$ of the elements of the ith-row of matrix ${\mathcal {M}}^{t-1}\widetilde{{\mathcal {M}}}$, i.e., of the ith-row of matrix $RD^{t-1}\widetilde{D}L$, since clearly ${\overline{a}}_s\le N$. $D^{t-1}\widetilde{D}$ is a diagonal matrix whose $\ell$th-diagonal element is given by:

$$\begin{aligned} \left( D^{t-1}\widetilde{D}\right) _{\ell }= & {} (2(1-r_d)r_\mathrm{div}(1-r_s)\lambda _{\ell }+(1-r_d)(1-r_\mathrm{div})(1-r_s))^{t-1} \\&\cdot (2(1-r_d)r_\mathrm{div}\lambda _{\ell }+(1-r_d)(1-r_\mathrm{div})) \\= & {} (1-r_s)^{t-1}(1-r_d)^t\left( 2\lambda _{\ell }r_\mathrm{div}+1-r_\mathrm{div}\right) ^t \end{aligned}$$

The result follows observing that: $\left( RD^{t-1}\widetilde{D}L\right) _{ik}=\sum _{\ell =0}^N\left( D^{t-1}\widetilde{D}\right) _{\ell }r_{i\ell }l_{\ell k}$. $\square$

Appendix F: Heuristic Proof of Proposition 6

Proposition

For all $t\in {\mathbb {N}}$ the value $r_s(t)$ which maximizes the expected number of selected B-cells at the tth maturation cycle is:

$$\begin{aligned} r_s(t)=\frac{1}{t} \end{aligned}$$

Hypothesis 1

${\mathcal {Q}}_N$ converges through its stationary distribution, denoted by $\mathbf {m}=(m_i)$, $i\,\in \,\{0,\dots ,N\}$.

Hypothesis 2

$Z_t$ explodes, where $(Z_t)_{t\in {\mathbb {N}}}$ is given by Definition 4.

Let ${\widetilde{Z}}_t$, $t\ge 0$ be the random variable describing the GC population size at time t before the selection mechanism is performed for this generation. For the sake of simplicity, let us suppose ${\widetilde{Z}}_0=1$. $({\widetilde{Z}}_t)_{t\in \mathbb {N}}$ is a MC on $\{0,1,2,\dots \}$. Denoted by $\tilde{p}_k:={\mathbb {P}}({\widetilde{Z}}_1=k)$, $k\,\in \,\{0,1,2\}$:

$$\begin{aligned} \left\{ \begin{array}{l} \tilde{p}_0=r_d \\ \tilde{p}_1=(1-r_d)(1-r_\mathrm{div}) \\ \tilde{p}_2=(1-r_d)r_\mathrm{div} \end{array}\right. \end{aligned}$$

(24)

It follows: $\tilde{m}:={\mathbb {E}}({\widetilde{Z}}_1)=(1-r_d)(1-r_\mathrm{div})+2(1-r_d)r_\mathrm{div}=(1-r_d)(1+r_\mathrm{div})$.

Conditioning to $Z_t=k$, ${\widetilde{Z}}_{t+1}$ is distributed as the sum of k independent copies of ${\widetilde{Z}}_1$, which gives:

$$\begin{aligned} {\mathbb {E}}({\widetilde{Z}}_t)={\mathbb {E}}(Z_{t-1}){\mathbb {E}}({\widetilde{Z}}_1)={\mathbb {E}}(Z_1)^{t-1}{\mathbb {E}}({\widetilde{Z}}_1)=(1-r_d)^t(1+r_\mathrm{div})^t(1-r_s)^{t-1} \end{aligned}$$

(25)

Thanks to Hypotheses 1 and 2, if t is big enough, there is approximately a proportion of $m_i$ elements in the ith-affinity class with respect to $\overline{\mathbf {x}}$. Therefore, on average at time t there are approximately $\sum _{i=0}^{{\overline{a}}_s}m_i{\mathbb {E}}({\widetilde{Z}}_t)$ B-cells in the GC belonging to an affinity class with index at most equal to ${\overline{a}}_s$ with respect to $\overline{\mathbf {x}}$, before the selection mechanism is performed for this generation. Each one of these cells can be submitted to selection with probability $r_s$, and in this case it will be positively selected. Hence:

$$\begin{aligned} {\mathbb {E}}(S_t)\simeq r_s\sum _{i=0}^{{\overline{a}}_s}m_i{\mathbb {E}}({\widetilde{Z}}_t)=(1-r_d)^t(1+r_\mathrm{div})^t(1-r_s)^{t-1}r_s\sum _{i=0}^{{\overline{a}}_s}m_i~, \end{aligned}$$

(26)

which is maximized at time $t\ge 1$ for $r_s(t)=1/t$.

Remark 8

One observes that the approximation in  (26) gives the same value for the optimal $r_s(t)$ as in Proposition 6. Nevertheless, it does not allow to describe exactly the behavior of ${\mathbb {E}}(S_t)$, since it is obtained by approximating the distribution of B-cells in the GC with their stationary distribution.