From clonal interference to Poissonian interacting trajectories

TL;DR

The paper introduces the Poissonian system of interacting trajectories (PIT) as a rigorous scaling limit for clonal interference in a Gerrish--Lenski regime of a Moran model with recurrent beneficial mutations. It proves joint convergence of log-frequencies, clonal fitnesses, average population fitness, and mutation ancestry to the PIT, and establishes a positive, finite speed of adaptation if the fitness increments have finite first moment (infinite otherwise), along with a functional central limit theorem for resident fitness. The PIT provides a tractable, stochastic-analytic framework that connects to classical Gerrish--Lenski heuristics while capturing the complex dynamics of competing mesoscopic types and their genealogies. The results offer a rigorous bridge between micro-level mutation events and macro-level adaptation rates, with potential extensions to general type spaces and alternative selection regimes, thereby enriching the theoretical understanding of clonal interference and rapid adaptation in large populations.

Abstract

We consider a population whose size $N$ is fixed over the generations, and in which random beneficial mutations arrive at a rate of order $1/\log N$ per generation. In this so-called Gerrish--Lenski regime, typically a finite number of contending mutations are present together with one resident type. These mutations compete for fixation, a phenomenon addressed as clonal interference. We introduce and study a Poissonian system of interacting trajectories (PIT), and prove that it arises as a large population scaling limit of the logarithmic sizes of the contending clonal subpopulations in a continuous-time Moran model with strong selection. We show that the PIT exhibits an almost surely positive asymptotic rate of fitness increase (speed of adaptation), which turns out to be finite if and only if fitness increments have a finite expectation. We relate this speed to heuristic predictions from the literature. Furthermore, we derive a functional central limit theorem for the fitness of the resident population in the PIT.

Figures (9)

• Figure 1: This figure depicts a simulation of a Moran model with mutation and selection (cf. Section \ref{['sec-model']}) in the Gerrish--Lenski regime with population size $N=500\,000$ and fitness increment distribution $\gamma = \frac{1}{2}\delta_{\{1\}}+\frac{1}{2}\delta_{\{2\}}$. Left: sub-population sizes divided by $N$, approximating logistic curves; middle: logarithmic sub-population sizes divided by $\log N$, approximately giving piecewise linear trajectories and making effects of clonal interference on the first (red) mutation visible; right: stylized version of these trajectories providing a good guess of the scaling limit -- i.e. the PIT.
• Figure 2: This is an example of a realisation of a PIT with $(T_1, \ldots, T_6) = (1.2,1.4,1.6,2.5,2.9,3.2)$. The Bernoulli variables $B_1$, …, $B_6$ have realisations $1,0,1,1,0,1$ and the initial slopes of $H_1,H_3,H_4,H_6$ are $(A_1,A_3, A_4, A_6) =(0.2, 1, 2, 1.6)$. Among $H_1, H_2, \dots$, the trajectory $H_3$ is the first one to reach height $1$. At the time $R_1$ at which this happens, the slope of $H_0$ jumps from $0$ to $-1$, the slope of $H_1$ jumps from $0.2$ to $-0.8$ and the slope of $H_4$ jumps from $1.5$ to $0.5$. The numbers in the top line of the figure are the current values of the resident fitness $F(t)$. In particular, $F(R_2) = 2$, and $F(R_3) = 2.6.$
• Figure 3: Illustration of the genealogy corresponding to Figure \ref{['fig-limitingexample']}. Dotted edges mark mutations lost to genetic drift, i.e. leading to non-contenders. Types 1, 3, 4 and 6 are contenders. Type 1 never becomes resident. Type 4 becomes resident but afterwards is "kinked to extinction" without having any descendant. Type 3, after becoming resident in a non-solitary way, spawns type 6 as its descendant, which then becomes resident in a solitary way. The bold arrows highlight mutations that will be present in the ancestral line of all future individuals. The depicted situation points to concepts that will be defined and discussed in Section \ref{['compfix']}.
• Figure 4: Illustration of Proposition \ref{['fixedk']}. Left: Moran model, $N=500\,000$, without mutations, with initial state $(X^N_1(0),\ldots,X^N_4(0)) = (N-\lfloor N^{0.8}\rfloor-\lfloor N^{0.3}\rfloor-\lfloor N^{0.1}\rfloor, \lfloor N^{0.8}\rfloor,\lfloor N^{0.3}\rfloor,\lfloor N^{0.1}\rfloor)$ and $(m_1,\ldots,m_4)=(0,0.8,1,1.5)$; horizontals: $1-\frac{\log\log N}{2\log N}$ resp. $\overline h_N=1-\frac{\log\log N}{3\log N}$; verticals mark $\tau_1^N, \sigma_1^N, \tau_2^N, \sigma_2^N$ from left to right. Right: Corresponding PIT $\mathbb H(((0.1,1.5),(0.3,1),(0.8,0.8),(1,0)),\varnothing)$; dotted verticals give $\tau_1=0.25$ and $\tau_2=1$ respectively.
• Figure 5: Illustration of part 1 of the proof of Lemma \ref{['clusterlength']}. Top: case $\{ S > T_i \}$. Between times $T_i-2/A_i$ and $T_i+2/A_i$ there is no birth time apart from $T_i$, while we have $V_i(T_i)=A_iB_i=2$. All slopes of trajectories that are still positive at time $T_i$ are at most $A_i/2$, and hence the $i$-th trajectory reaches height 1 at time $R \leq T_i + 2/A_i$ at latest, kinking all other trajectories with current heights in $(0,1]$ to a negative slope. In the picture, the only trajectory still having a positive slope at time $T_i$ is the brown one, and $S$ is the time when this trajectory reaches height 1. The slope of the brown trajectory in $[T_i,S)$ equals $0.5$, and thus at time $R$, the blue trajectory is kinked to slope $2-0.5=1.5 \geq A_i/2$. The gray trajectory corresponds to the mutant who is resident at time $T_i$ (this is the last resident before the brown one). Bottom: case $\{ S=T_i \}$. Now the brown mutant is absent, so that the blue trajectory suffers no kink before reaching height 1, and the previous resident before the blue one is the gray one. Note that here, the time when the blue trajectory reaches height 1 is $R=T_i+1/A_i$.

Theorems & Definitions (52)

• Definition 2.1: Dynamics of the PIT
• Remark 2.2: Discarding the trajectories of initial slope $0$
• Lemma 2.3
• proof
• Definition 2.4: Resident type, fitness of types, resident fitness, resident change times
• Lemma 2.5
• Definition 2.6: Genealogy of mutations in the PIT
• Theorem 2.7
• Theorem 2.8
• Definition 2.9