# A geometric speed limit for acceleration by natural selection in
evolutionary processes

Masahiro Hoshino<EMAIL_ADDRESS>Department of
Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan  Ryuna Nagayama<EMAIL_ADDRESS>Department of
Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan  Universal Biology Institute, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan  Kohei Yoshimura  Department of Physics, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Universal
Biology Institute, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan  Jumpei F. Yamagishi  Universal Biology Institute, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Graduate
School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku,
Tokyo 153-8902, Japan  Sosuke Ito  Department of Physics, The University of
Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Universal Biology
Institute, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan

###### Abstract

We derived a new speed limit in population dynamics, which is a fundamental
limit on the evolutionary rate. By splitting the contributions of selection
and mutation to the evolutionary rate, we obtained the new bound on the speed
of arbitrary observables, named the selection bound, that can be tighter than
the conventional Cramér–Rao bound. Remarkably, the selection bound can be much
tighter if the contribution of selection is more dominant than that of
mutation. This tightness can be geometrically characterized by the correlation
between the observable of interest and the growth rate. We also numerically
illustrate the effectiveness of the selection bound in the transient dynamics
of evolutionary processes and discuss how to test our speed limit
experimentally.

Introduction.— Biological populations fluctuate through natural selection and
mutation due to various environmental influences. While mutation increases
their diversity, natural selection increases the fraction of highly adaptive
traits in the population. This competition between selection and mutation
leads to evolution Kimura (1983); Ohta (1992); Frank (2019). Recent
improvements in experimental methods have enabled researchers to
quantitatively observe the evolutionary dynamics of actual biological
communities Arjan G  _et al._ (1999); Wakamoto  _et al._ (2005); Barrick  _et
al._ (2009); De Visser and Krug (2014); Hashimoto  _et al._ (2016); Baym  _et
al._ (2016); De Martino  _et al._ (2016); Good  _et al._ (2017); Lukačišinová
and Bollenbach (2017); Furusawa  _et al._ (2018); Van den Bergh  _et al._
(2018). For example, Ref. Baym  _et al._ (2016) visualized how selection and
mutation together influence the adaptation dynamics of a bacterial
population’s growth.

Though these recent experiments allow us to measure the evolutionary rate
quantitatively, the classical theories for evolution were not sufficiently
quantitative. For example, the principal ideas of evolution, such as natural
selection in Darwinian evolution Darwin (2004), have not been clearly
expressed quantitatively. A famous theorem on the evolutionary rate known as
Fisher’s fundamental theorem of natural selection Fisher (1958); Ewens (1989);
Frank (1997); Baez (2021), which claims a relation between the increment of
the mean fitness and the fitness variance, has also been misunderstood by many
researchers because it is given in a quantitatively vague expression Frank and
Slatkin (1992). One exception is the Price equation Price (1972); Frank and
Slatkin (1992); Grafen (2000); Frank (2012a); Baum (2017); Frank (2018); Frank
and Bruggeman (2020), which provides a clear-cut relation between the
observables associated with traits and their fitness. Because the Price
equation is a purely mathematical relation based on identity, we need to
consider specific population dynamics Malthus (1872); Leibler and Kussell
(2010); Crow and Kimura (2017); Basener and Sanford (2018) to identify its
physical implication for the evolutionary rate.

Recently, quantitative theoretical approaches have been developed by analogy
with another developing field of stochastic thermodynamics Sekimoto (2010);
Seifert (2012). In population dynamics models such as the Lotka–Volterra model
Strogatz (2018) and the lineage trees Wakamoto  _et al._ (2012); Lambert and
Kussell (2015); Hoffmann and Olivier (2016); Nozoe  _et al._ (2017), several
quantitative inequalities or trade-off relations for the evolutionary
processes have been investigated Andrae  _et al._ (2010); Qian (2014);
Kobayashi and Sughiyama (2015); Sughiyama  _et al._ (2015); Sughiyama and
Kobayashi (2017); Kobayashi and Sughiyama (2017); García-García  _et al._
(2019); Genthon and Lacoste (2020, 2021); Yoshimura and Ito (2021a);
Kolchinsky (2021) by analogy with thermodynamic laws such as the second law of
thermodynamics Esposito and Van den Broeck (2010); Van den Broeck and Esposito
(2010) and thermodynamic uncertainty relations Barato and Seifert (2015). As a
notable result, the evolutionary rate has been discussed quantitatively in
Ref. Zhang  _et al._ (2020) by applying the information-geometric speed limits
Crooks (2007); Ito (2018). The speed limits have been discussed as a classical
counterpart of the quantum speed limits Mandelstam and Tamm (1991); Anandan
and Aharonov (1990); Margolus and Levitin (1998); Taddei  _et al._ (2013);
Pires  _et al._ (2016); Shanahan  _et al._ (2018); Okuyama and Ohzeki (2018)
in the context of a connection between information geometry Amari (2016) and
stochastic thermodynamics. As a constraint on the speed of dynamics, the speed
limits offer a basis to discuss the evolutionary rate quantitatively. These
speed limits have also been generalized to the speed of observable Ito and
Dechant (2020); Nicholson  _et al._ (2020); Yoshimura and Ito (2021b); Ito
(2022); Ashida  _et al._ (2021) based on the Cramér–Rao bound Amari (2016);
Rao (1992), well known in information geometry. This information-geometric
approach would be promising as a quantitative theory for the evolutionary rate
because of a deep connection between the Cramér–Rao bound and the Price
equation Frank (2018); Frank and Bruggeman (2020) and because this approach
may be compatible with the existing information-theoretic and stochastic
methods for evolutionary dynamics Sato  _et al._ (2003); Michel  _et al._
(2005); Wolf  _et al._ (2005); Kussell and Leibler (2005); Rivoire and Leibler
(2011); Frank (2012b, 2013); Kaneko  _et al._ (2015); Xue and Leibler (2018);
Nakashima and Kobayashi (2022). Indeed, several applications and
generalizations of the speed limits have been recently studied to understand
the speed in population dynamics quantitatively Adachi  _et al._ (2022);
García-Pintos (2022).

However, those previous studies Zhang  _et al._ (2020); Adachi  _et al._
(2022); García-Pintos (2022) did not focus on the competing situations of
natural selection and mutation, even though selection and mutation together
shape evolution. Here, we pose the following unresolved issue: how and when
does the evolutionary rate change in the competing situations of natural
selection and mutation? This question would be crucial for a quantitative
understanding of evolutionary processes where the competition between
selection and mutation can enhance the evolutionary rate, as observed in Ref.
Baym  _et al._ (2016).

To resolve such an issue, we theoretically evaluated the evolutionary rate by
decomposing it into the contributions of natural selection and mutation in
population dynamics, thereby deriving a new speed limit. This speed limit is
tighter than the conventional Cramér–Rao bound when natural selection is
dominant compared to mutation (e.g., in transient dynamics of evolution), as
analytically proven and numerically illustrated. It describes how natural
selection accelerates evolution.

Setup.— To discuss speed limits for observables in population dynamics, we
consider a model consisting of selection and mutation between multiple traits
Crow and Kimura (2017) (see also Fig. 1(a)). Suppose a population consists of
subpopulations with $n$ different traits, and $N_{i}(t)$ and $\lambda_{i}$
denote the number and growth rate of individuals in the subpopulation with the
$i$-th trait at time $t$, respectively. The traits may be phenotypic,
genotypic, or epigenetic properties. We denote the vector of growth rates
simply as $\lambda\,{=}\,(\lambda_{i})$. Mutation is assumed to be a Markovian
process with transition rate matrix $\mathsf{W}\,{=}\,(W_{ij})$, where the
$(i,j)$-element $W_{ij}$ indicates the transition rate from trait $j$ to $i$
if $i\,{\neq}\,j$, and the elements satisfy $\sum_{i=1}^{n}W_{ij}{=}\,0$ and
$W_{ij}\,{\geq}\,0\;(i\,{\neq}\,j)$. We assume that $\lambda$ and $\mathsf{W}$
are time-independent because we consider a stationary environment. Then
$N_{i}(t)$ follows the following differential equation

$\displaystyle\frac{d}{dt}N_{i}(t)=\lambda_{i}N_{i}(t)+\sum_{j=1}^{n}W_{ij}N_{j}(t).$
(1)

The first term on the right-hand side represents the change in the population
due to selection and the second term represents the change in the population
due to mutation. With $N_{\rm tot}(t)\,{\coloneqq}\,\sum_{i=1}^{n}N_{i}(t)$,
the proportion $p_{i}(t)\,{\coloneqq}\,N_{i}(t)/N_{\rm tot}(t)$ of each
subpopulation satisfies the definition of the probability distribution, i.e.,
the non-negativity $p_{i}(t)\,{\geq}\,0$ and the normalization
$\sum_{i=1}^{n}p_{i}(t)\,{=}\,1$. From Eq. (1), this “probability
distribution” follows a nonlinear master equation,

$\displaystyle\frac{d}{dt}p_{i}(t)=\Delta\lambda_{i}p_{i}(t)+\sum_{j=1}^{n}W_{ij}p_{j}(t),$
(2)

where the ensemble average of an observable $A\,{=}\,\\{A_{i}\\}_{i=1}^{n}$
with respect to $p_{i}(t)$ is defined as $\langle
A\rangle\,{\coloneqq}\,\sum_{i=1}^{n}p_{i}(t)A_{i}$ and $\Delta
A_{i}\,{=}\,A_{i}-\langle A\rangle$ denotes the deviation of $A_{i}$.

Figure 1: (a) Schematic illustration of the model. Population size changes due
to selection on growth and mutation. (b) Schematic illustration of the
selection bound. Compared to the case where the population changes only due to
mutation, the change of $\langle R\rangle$ can be accelerated when both
selection and mutation affect the population dynamics.

Speed limit and information geometry.— We here briefly explain the
conventional information-geometric speed limit for a time-independent
observable $R\,{=}\,\\{R_{i}\\}_{i=1}^{n}$. The speed of observable $R$ is
defined as

$\displaystyle v_{R}\coloneqq\frac{1}{\sqrt{\mathrm{Var}[R]}}\frac{d\langle
R\rangle}{dt},$ (3)

where $\mathrm{Var}[R]\,{\coloneqq}\,\langle(\Delta R)^{2}\rangle$ is the
variance of $R$. In population dynamics, $v_{R}$ quantifies the evolutionary
rate with respect to observable $R$. For example, the evolutionary rate with
respect to the growth rate, $v_{\lambda}$, is given by the time derivative of
the averaged growth rate $d\langle\lambda\rangle/dt$ normalized by its
standard deviation $\sqrt{\mathrm{Var}[\lambda]}$. The speed limit for
observable, known as the Cramér–Rao bound, is a universal constraint on this
speed $v_{R}$ for any observable $R$ Ito and Dechant (2020); Nicholson  _et
al._ (2020):

$\displaystyle-v_{\rm info}\leq v_{R}\leq v_{\rm info},$ (4)

which holds for arbitrary dynamics of a probability distribution 111We remark
that the Cramér–Rao bound holds even for time-dependent observables by
changing the definition of $v_{R}$ as $v_{R}\,{=}\,(d_{t}\langle
R\rangle-\langle d_{t}R\rangle)/\sqrt{\mathrm{Var}[R]}$.. Here $v_{\rm info}$
is defined as the square root of the Fisher information Cover (1999); Frank
(2009); Ito and Dechant (2020):

$\displaystyle v_{\rm
info}\coloneqq\sqrt{\sum_{i=1}^{n}p_{i}\quantity(\frac{d\ln p_{i}}{dt})^{2}}.$
(5)

In information geometry, we can interpret it as the speed of a probability
distribution moving on a manifold of distributions.

Fitness and Price equation.— The square root of the Fisher information $v_{\rm
info}$ not only indicates the speed of the probability distribution but also
characterizes the population dynamics because it is identified with the
variance of fitness Frank (2018); Baez (2021). We here introduce the fitness
$f_{i}$ of trait $i$ as the effective growth rate of $N_{i}$:

$\displaystyle f_{i}\coloneqq\frac{d}{dt}\ln N_{i}(t).$ (6)

The ensemble average of the fitness is equal to the effective growth rate of
the total population: $\langle f\rangle\,{=}\,d\ln N_{\rm tot}/dt$. Together
with Eq. (5) and $\ln p_{i}(t)\,{=}\,\ln N_{i}(t)-\ln N_{\rm tot}(t)$, these
relations lead to the equality

$\displaystyle v_{\rm info}=\sqrt{\sum_{i=1}^{n}p_{i}\quantity(\Delta
f_{i})^{2}}=\sqrt{\mathrm{Var}[f]}.$ (7)

That is, $v_{\rm info}$ also quantifies the diversity of each trait’s fitness.
Accordingly, Eq. (4) implies that the variance of fitness limits the speed of
an arbitrary observable.

On the other hand, we can discuss the role of $v_{R}$ in population dynamics
based on the Price equation Price (1972); Frank and Slatkin (1992); Grafen
(2000); Baum (2017); Frank (2018); Frank and Bruggeman (2020); Frank (2012a).
A special case of the Price equation for a time-independent observable
provides a connection with the time derivative of the stochastic entropy in
the system, $\dot{\sigma}_{i}(t)\,{\coloneqq}\,-d\ln p_{i}(t)/dt$, as

$\displaystyle\frac{d\langle R\rangle}{dt}=\mathrm{Cov}[R,-\dot{\sigma}],$ (8)

where the covariance of two observables is defined as
$\mathrm{Cov}[A,B]\,{\coloneqq}\,\langle\Delta A\Delta B\rangle$. This
equation indicates that the evolutionary rate is governed by the stochastic
entropy change rate in the system. We remark that $\dot{\sigma}$ is directly
connected to the fitness as $-\dot{\sigma}_{i}\,{=}\,\Delta f_{i}$, so that
its ensemble average and variance satisfy $\langle\dot{\sigma}\rangle\,{=}\,0$
and
$\sqrt{\mathrm{Var}[\dot{\sigma}}]\,{=}\,\sqrt{\mathrm{Var}[f]}\,{=}\,v_{\rm
info}$. From Eq. (8), $v_{R}$ is rewritten as

$\displaystyle
v_{R}=\frac{\mathrm{Cov}[R,-\dot{\sigma}]}{\sqrt{\mathrm{Var}[R]}}=\frac{\mathrm{Cov}[R,f]}{\sqrt{\mathrm{Var}[R]}},$
(9)

which implies that the speed of an observable can be interpreted in terms of
the covariance between the observable and the fitness. From Eqs. (7) and (9),
the Cramér–Rao bound (4) can be derived by applying the Cauchy–Schwarz
inequality
$-\sqrt{\mathrm{Var}[R]}\sqrt{\mathrm{Var}[f]}\,{\leq}\,\mathrm{Cov}[R,f]\,{\leq}\,\sqrt{\mathrm{Var}[R]}\sqrt{\mathrm{Var}[f]}$.

Main result: Selection bound.— We explain the main result which is a new speed
limit based on the contribution of selection in evolutionary dynamics. The key
idea for the main result is the decomposition of the stochastic entropy change
rate in the system. In the population dynamics model (2), $\dot{\sigma}$ can
be decomposed into two parts as

$\displaystyle\dot{\sigma}={\dot{\sigma}}^{\lambda}+{\dot{\sigma}}^{\mathsf{W}},$
(10)

where ${\dot{\sigma}}^{\lambda}_{i}\,{\coloneqq}\,-\Delta\lambda_{i}$ and
${\dot{\sigma}}^{\mathsf{W}}_{i}\,{\coloneqq}\,-\sum_{j=1}^{n}W_{ij}p_{j}/p_{i}$
are the stochastic entropy change rate in the system due to only selection and
mutation, respectively. We remark that ${\dot{\sigma}}^{\mathsf{W}}$ is
rewritten as
${\dot{\sigma}}^{\mathsf{W}}_{i}\,{=}\,-\Delta(f_{i}-\lambda_{i})$. These
quantities also satisfy
$\langle\dot{\sigma}^{\lambda}\rangle\,{=}\,\langle\dot{\sigma}^{\mathsf{W}}\rangle\,{=}\,0$,
as $\dot{\sigma}$ does. Considering this decomposition, we introduce the
following quantities,

$\begin{split}v_{R}^{\lambda}\coloneqq\dfrac{\mathrm{Cov}[R,-\dot{\sigma}^{\lambda}]}{\sqrt{\mathrm{Var}[R]}},\quad&v_{\rm
info}^{\lambda}\coloneqq\sqrt{\mathrm{Var}[\dot{\sigma}^{\lambda}]},\\\
v_{R}^{\mathsf{W}}\coloneqq\dfrac{\mathrm{Cov}[R,-\dot{\sigma}^{\mathsf{W}}]}{\sqrt{\mathrm{Var}[R]}},\quad&v_{\rm
info}^{\mathsf{W}}\coloneqq\sqrt{\mathrm{Var}[\dot{\sigma}^{\mathsf{W}}]},\end{split}$
(11)

where the upper two can be interpreted as $v_{R}$ and $v_{\rm info}$ without
the contribution of mutation, while the lower ones are interpreted as those
without selection. In other words, the former are speeds stemming solely from
selection, while the latter mutation only. These quantities are given by the
variance and covariance of the measurable observables $R$, $\lambda$, and
$f-\lambda$;
$v_{R}^{\lambda}\,{=}\,\mathrm{Cov}[R,\lambda]/\sqrt{\mathrm{Var}[R]}$,
$v_{\rm info}^{\lambda}\,{=}\,\sqrt{\mathrm{Var}[\lambda]}$,
$v_{R}^{\mathsf{W}}\,{=}\,\mathrm{Cov}[R,f-\lambda]/\sqrt{\mathrm{Var}[R]}$,
and $v_{\rm info}^{\mathsf{W}}\,{=}\,\sqrt{\mathrm{Var}[f-\lambda]}$. By
decomposing $v_{R}$ into the contributions of selection and mutation, we
obtain a new speed limit:

$\displaystyle v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}}\leq v_{R}\leq
v_{R}^{\lambda}+v_{\rm info}^{\mathsf{W}}.$ (12)

We call this new speed limit the selection bound because the speed of
observable $v_{R}$ is accelerated by the effect of selection
$v_{R}^{\lambda}$, compared to the case where no selection occurs ($-v_{\rm
info}^{\mathsf{W}}\,{\leq}\,v_{R}\,{\leq}\,v_{\rm info}^{\mathsf{W}}$ for
$\lambda\,{=}\,0$). Therefore, this bound quantifies the acceleration of the
evolutionary rate by natural selection compared to mutational dynamics in the
absence of the selection (see also Fig. 1(b)).

The equation (12) is derived essentially from the Cauchy–Schwarz inequality as
well as the Cramér–Rao bound. To simplify its derivation, we define an inner
product and the associated norm for observables as $\langle
A,B\rangle\,{\coloneqq}\,\sum_{i=1}^{n}p_{i}A_{i}B_{i}$ and
$\|A\|\,{:=}\,\sqrt{\langle A,A\rangle}$, respectively. We can rewrite $v_{R}$
as

$\displaystyle v_{R}$ $\displaystyle=\dfrac{\langle\Delta
R,\Delta\lambda+\Delta(f-\lambda)\rangle}{\|\Delta
R\|}=v_{R}^{\lambda}+v_{R}^{\mathsf{W}}.$ (13)

Applying the Cauchy–Schwarz inequality $-\|\Delta
R\|\|\Delta(f-\lambda)\|\,{\leq}\,\langle\Delta
R,\Delta(f-\lambda)\rangle\,{\leq}\,\|\Delta R\|\|\Delta(f-\lambda)\|$ to Eq.
(13), we can obtain the selection bound.

This inner product also provides a useful geometric interpretation to discuss
the effectiveness of the selection bound (see also Fig. 2(a)). In the
geometric interpretation, $v_{\mathrm{info}}\,{=}\,\|\dot{\sigma}\|$,
$v_{\mathrm{info}}^{\lambda}\,{=}\,\|\dot{\sigma}^{\lambda}\|$ and
$v_{\mathrm{info}}^{\mathsf{W}}\,{=}\,\|\dot{\sigma}^{\mathsf{W}}\|$ are the
norms of $-\dot{\sigma}$, $-\dot{\sigma}^{\lambda}$ and
$-\dot{\sigma}^{\mathsf{W}}$, respectively, and thus the triangle inequality
$\absolutevalue{v_{\rm info}^{\lambda}-v_{\rm
info}^{\mathsf{W}}}\,{\leq}\,v_{\rm info}\,{\leq}\,v_{\rm
info}^{\lambda}+v_{\rm info}^{\mathsf{W}}$ holds. In addition,
$v_{R}\,{=}\,\langle\Delta R,-\dot{\sigma}\rangle/\|\Delta R\|$,
$v_{R}^{\lambda}\,{=}\,\langle\Delta R,-\dot{\sigma}^{\lambda}\rangle/\|\Delta
R\|$, and $v_{R}^{\mathsf{W}}\,{=}\,\langle\Delta
R,-\dot{\sigma}^{\mathsf{W}}\rangle/\|\Delta R\|$ are the norms of the
projections of $-\dot{\sigma}$, $-\dot{\sigma}^{\lambda}$ and
$-\dot{\sigma}^{\mathsf{W}}$ onto $\Delta R$, respectively. Then, the angle
between $\Delta R$ and $-\dot{\sigma}^{\lambda}\,{=}\,\Delta\lambda$ can be
defined as

$\displaystyle\theta_{R}^{\lambda}\coloneqq\arccos(\frac{\langle\Delta
R,\Delta\lambda\rangle}{\|\Delta
R\|\|\Delta\lambda\|})=\arccos(\frac{v_{R}^{\lambda}}{v_{\rm
info}^{\lambda}}),$ (14)

or equivalently $v_{R}^{\lambda}\,{=}\,v_{\rm
info}^{\lambda}\cos\theta_{R}^{\lambda}$. It quantifies the strength of the
correlation between $R$ and $\lambda$ 222We remark that a similar concept of
the angle can be seen in the Price equation Frank (2012a).. If there is a
positive correlation between an observable $R$ and the growth rate $\lambda$
(i.e., $\cos\theta_{R}^{\lambda}\,{\geq}\,0$), $v_{R}^{\lambda}$ is positive
and both upper and lower bounds in the selection bound shift to the positive
direction (i.e., $\pm v_{\rm info}^{\mathsf{W}}+v_{R}^{\lambda}$), compared to
the case without selection (i.e., $\pm v_{\rm info}^{\mathsf{W}}$). Therefore,
positive correlations between observables and the growth rate can lead to
faster evolution. From a biological viewpoint, it indicates that if the value
of the observable $R$ tends to be larger in fast-growing traits, its
evolutionary rate can be accelerated, and vice versa. Noting the relation
$v_{R}^{\lambda}\,{=}\,v_{\rm info}^{\lambda}\cos\theta_{R}^{\lambda}$, not
only stronger correlations between observables and growth rate (i.e., larger
$\cos\theta_{R}^{\lambda}$) but also greater contributions of selection (i.e.,
larger $v_{\rm info}^{\lambda}$, as discussed above) together allow a faster
evolutionary rate.

Figure 2: (a) Relations between the quantities presented in this Letter in the
inner product space. $-\dot{\sigma}$ can be written by the sum of the
contributions of selection and mutation, $-{\dot{\sigma}}^{\lambda}$ and
$-{\dot{\sigma}}^{\mathsf{W}}$. The norms of
$-\dot{\sigma},-{\dot{\sigma}}^{\lambda},-{\dot{\sigma}}^{\mathsf{W}}$ are
$v_{\rm info},v_{\rm info}^{\lambda},v_{\rm info}^{\mathsf{W}}$, respectively.
The speed $v_{R}$ can be expressed by the projection of $-\dot{\sigma}$ in the
direction of $\Delta R$, while the projections of $-{\dot{\sigma}}^{\lambda}$
and $-{\dot{\sigma}}^{\mathsf{W}}$ are $v_{R}^{\lambda}$ and
$v_{R}^{\mathsf{W}}$, respectively. Thus,
$v_{R}=v_{R}^{\lambda}+v_{R}^{\mathsf{W}}$ holds. (b) The range of
$\theta_{R}^{\lambda}$ that determines whether the selection bound or the
Cramér–Rao bound evaluates $v_{R}$ tightly or loosely. If
$\theta^{\ast}\,{\leq}\,\theta_{R}^{\lambda}\,{\leq}\,\pi$, the selection
bound gives a tighter upper bound, and if
$0\,{\leq}\,\theta_{R}^{\lambda}\,{\leq}\,\pi-\theta^{\ast}$, the selection
bound gives a tighter lower bound. Therefore, both upper and lower bounds are
tight when $\theta_{R}^{\lambda}$ is located in the orange area
$[\theta^{\ast},\pi-\theta^{\ast}]$.

Finally, let us compare our bound (12) with the conventional Cramér–Rao bound
(4) to see how it quantifies the competition between selection and mutation.
To this end, we define another angle $\theta^{\ast}$ as

$\displaystyle\theta^{\ast}\coloneqq\arccos{\quantity(\dfrac{v_{\rm
info}-v_{\rm info}^{\mathsf{W}}}{v_{\rm info}^{\lambda}})},$ (15)

which is well-defined because the argument of the arccosine is always in
$[-1,1]$ from the triangle inequality. Using $\theta_{R}^{\lambda}$ and
$\theta^{\ast}$, the following case separation gives the condition in which
case the Cramér–Rao bound or the selection bound gives better evaluation:

$\displaystyle\begin{cases}0\leq\theta_{R}^{\lambda}\leq\theta^{\ast}&\implies
v_{R}\leq v_{\rm info}\leq v_{R}^{\lambda}+v_{\rm info}^{\mathsf{W}}\\\
\theta^{\ast}\leq\theta_{R}^{\lambda}\leq\pi&\implies v_{R}\leq
v_{R}^{\lambda}+v_{\rm info}^{\mathsf{W}}\leq v_{\rm info}\\\
0\leq\theta_{R}^{\lambda}\leq\pi-\theta^{\ast}&\implies-v_{\rm info}\leq
v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}}\leq v_{R}\\\
\pi-\theta^{\ast}\leq\theta_{R}^{\lambda}\leq\pi&\implies
v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}}\leq-v_{\rm info}\leq
v_{R}\end{cases}.$ (16)

This implies that if $\theta^{\ast}$ is smaller than $\pi/2$ and
$\theta_{R}^{\lambda}$ is in the range $[\theta^{\ast},\pi-\theta^{\ast}]$,
then the selection bound will bound $v_{R}$ more tightly than the Cramér–Rao
bound, both lower and above (see Fig. 2(b)). Since this range
$[\theta^{\ast},\pi-\theta^{\ast}]$ does not depend on the choice of specific
observables, we can discuss the tightness of the selection bound
quantitatively only by the angles. Because $\theta^{\ast}$ is given as the
arccosine of the ratio between
$v_{\mathrm{info}}-v_{\mathrm{info}}^{\mathsf{W}}$ and
$v_{\mathrm{info}}^{\lambda}$ and $v_{\rm info}$ is given as the sum of
$v_{\rm info}^{\lambda}$, $v_{\rm info}^{\mathsf{W}}$ and a correlation term
between them (cf. cosine theorem), $\theta^{\ast}$ becomes smaller when the
contribution of selection $v_{\rm info}^{\lambda}$ gets larger, since the
ratio gets closer to one. That is, if the contribution of selection is larger
than mutation ($v_{\mathrm{info}}^{\lambda}\gg
v_{\mathrm{info}}^{\mathsf{W}}$), the range becomes wider so that the
selection bound can give a better bound on $v_{R}$ for a wider variety of
observables $R$. Such a tendency is indeed observed in the numerical
calculations below.

Example.— We illustrate our results by numerical calculations (Fig. 3). To
consider a situation where the contribution of selection is dominant, we have
the parameters in (2), $\\{\lambda_{i}\\}$ and $\\{W_{ij}\\}_{i\neq j}$,
uniformly sampled from $[-\lambda_{\rm max},\lambda_{\rm max}]$ and
$[0,\mathsf{W}_{\rm max}]$, and set $\lambda_{\rm max}/\mathsf{W}_{\rm
max}=100$. Given that the present results hold for arbitrary time-independent
observables, we also uniformly sample $\\{R_{i}\\}$ within the range
$[-10,10]$ rather than taking a specific observable. In Fig. 3, at an early
stage of the evolutionary dynamics, or far from the steady state, the
selection bound better restricts $v_{R}$. From the ecological perspective,
this behavior seems reasonable: selection dominantly contributes to the
evolutionary processes far from steady states because beneficial mutation gets
less likely as evolution progresses. As proven above, the selection bound is
tighter than the Cramér–Rao bound when $\theta_{R}^{\lambda}$ is in the range
$[\theta^{\ast},\pi-\theta^{\ast}]$ (colored in orange in Fig. 3). This range
is wider when the selection is dominant. A more precise evaluation of the
speed limits is discussed in Supplemental Material (SM) Sup .

Complementary result: Mutation bound.— The discussion so far has focused on
how the evolutionary rate is accelerated by selection on growth. On the other
hand, we can discuss acceleration by mutation by inverting the roles of
$\mathsf{W}$ and $\lambda$ in the selection bound. Concretely, we can derive
the bound

$\displaystyle v_{R}^{\mathsf{W}}-v_{\rm info}^{\lambda}\leq v_{R}\leq
v_{R}^{\mathsf{W}}+v_{\rm info}^{\lambda}.$ (17)

We call this bound the mutation bound because it extracts the effect of
mutation $v_{R}^{\mathsf{W}}$. The same analysis can be performed for the
mutation bound as for the selection bound. With both the selection bound and
the mutation bound, we can better capture the characteristics of evolutionary
processes, especially in the competing situation of natural selection and
mutation (see SM Sup  for details).

Figure 3: Numerical calculation for the conventional speed limit (4) by the
Cramér–Rao bound and the new speed limit (12) by the selection bound. The
horizontal axis is time, and $v_{R},\;v_{R}^{\mathsf{W}}+v_{\rm
info}^{\lambda},\;v_{R}^{\mathsf{W}}-v_{\rm info}^{\lambda},\;v_{\rm info}$,
and $-v_{\rm info}$ are plotted in the upper row. The lower row shows the
angles, $\theta_{R}^{\lambda},\;\theta^{\ast},$ and $\pi-\theta^{\ast}$.

Experimental accessibility.—Our speed limit is quantitatively testable by
actual experiments using single-cell lineage tree data. Recent advances in
experimental techniques enable us to measure when and into what each cell
mutates or divides. For example, by analyzing time-lapse images of growing
bacteria Nozoe  _et al._ (2017), we can measure transitions in phenotypic
traits (e.g., cell sizes, shapes, and intracellular concentration of a
particular protein) and proliferation dynamics at the same time. The number of
individuals with $i$-th trait at time $t$ that have experienced $K$ divisions
since time $t^{\prime}$, denoted as $N_{i}(K,t;t^{\prime})$, can be obtained
in such an experiment. This quantity $N_{i}(K,t;t^{\prime})$ enables us to
compute the instantaneous values of all the quantities in our speed limit,
$p_{i}(t)$, ${\dot{\sigma}}_{i}^{\lambda}$, ${\dot{\sigma}}_{i}^{\mathsf{W}}$,
${\dot{\sigma}}_{i}^{\lambda}$, $v_{R}^{\lambda},v_{R}^{\mathsf{W}},v_{\rm
info}^{\lambda}$, and $v_{\rm info}^{\mathsf{W}}$ (see SM Sup ). Therefore, we
can experimentally check the tightness of our speed limit for arbitrary
observable $R$, which quantifies the contribution of observable $R$ to the
evolutionary rate in the selection process.

Conclusion.— We derived a novel speed limit, the selection bound, that
considers the contributions of the selection and mutation separately when the
competition between selection and mutation exists. The core of this result is
the decomposition of the stochastic entropy change rate into the selection and
mutation part. It allows us to understand the limitations of the speeds of
observables more precisely than the conventional speed limit from the
Cramér–Rao bound. Although the limits from the selection bound depend on the
observables we consider, the “tendency” for the selection bound to give a
better bound than the Cramér–Rao bound only depends on the selection strength.
The selection bound should be effective in selection-dominant situations such
as environmental shift conditions.

The decomposition of the stochastic entropy change rate in this Letter may be
applicable to other nonlinear dynamics. For example, the generalized Lindblad
equation for post-selection in quantum dynamics Wiseman (1994); Cresser  _et
al._ (2006); Zhou (2022) has a similar nonlinear term originated by the
normalization of a probability distribution. Our decomposition into a
nonlinear contribution (i.e., selection) and a linear contribution (i.e.,
mutation) may be generalized for such an equation to derive a specialized
speed limit that characterizes the property of post-selection.

###### Acknowledgements.

M. H., J. F. Y., and S. I thank Shion Orii for valuable discussions. R. N. and
S. I. thank Chikara Furusawa and Yusuke Himekoka for helpful comments. S. I.
and J. F. Y. also thank Kouki Yamada for valuable discussions. S. I. is
supported by JSPS KAKENHI Grants No. 19H05796, No. 21H01560, and No. 22H01141,
JST Presto Grant No. JPMJPR18M2, and UTEC-UTokyo FSI Research Grant Program.
K. Y. and J. F. Y. are supported by Grant-in-Aid for JSPS Fellows (Grant No.
22J21619 and No. 21J22920, respectively). M. H. and R. N. contributed equally
to this work.

## References

  * Kimura (1983) M. Kimura,  _The neutral theory of molecular evolution_ (Cambridge University Press, 1983).
  * Ohta (1992) T. Ohta, Annual review of ecology and systematics , 263 (1992).
  * Frank (2019) S. A. Frank, in  _Foundations of Social Evolution_ (Princeton University Press, 2019).
  * Arjan G  _et al._ (1999) J. Arjan G, M. d. Visser, C. W. Zeyl, P. J. Gerrish, J. L. Blanchard, and R. E. Lenski, Science 283, 404 (1999).
  * Wakamoto  _et al._ (2005) Y. Wakamoto, J. Ramsden, and K. Yasuda, Analyst 130, 311 (2005).
  * Barrick  _et al._ (2009) J. E. Barrick, D. S. Yu, S. H. Yoon, H. Jeong, T. K. Oh, D. Schneider, R. E. Lenski, and J. F. Kim, Nature 461, 1243 (2009).
  * De Visser and Krug (2014) J. De Visser and J. Krug, Nature Reviews Genetics 15, 480 (2014).
  * Hashimoto  _et al._ (2016) M. Hashimoto, T. Nozoe, H. Nakaoka, R. Okura, S. Akiyoshi, K. Kaneko, E. Kussell, and Y. Wakamoto, Proceedings of the National Academy of Sciences 113, 3251 (2016).
  * Baym  _et al._ (2016) M. Baym, T. D. Lieberman, E. D. Kelsic, R. Chait, R. Gross, I. Yelin, and R. Kishony, Science 353, 1147 (2016).
  * De Martino  _et al._ (2016) D. De Martino, F. Capuani, and A. De Martino, Physical biology 13, 036005 (2016).
  * Good  _et al._ (2017) B. H. Good, M. J. McDonald, J. E. Barrick, R. E. Lenski, and M. M. Desai, Nature 551, 45 (2017).
  * Lukačišinová and Bollenbach (2017) M. Lukačišinová and T. Bollenbach, Current Opinion in Biotechnology 46, 90 (2017).
  * Furusawa  _et al._ (2018) C. Furusawa, T. Horinouchi, and T. Maeda, Current opinion in biotechnology 54, 45 (2018).
  * Van den Bergh  _et al._ (2018) B. Van den Bergh, T. Swings, M. Fauvart, and J. Michiels, Microbiology and Molecular Biology Reviews 82, e00008 (2018).
  * Darwin (2004) C. Darwin,  _On the origin of species, 1859_ (Routledge, 2004).
  * Fisher (1958) R. A. Fisher,  _The genetical theory of natural selection_ (Рипол Классик, 1958).
  * Ewens (1989) W. J. Ewens, Theoretical population biology 36, 167 (1989).
  * Frank (1997) S. A. Frank, Evolution 51, 1712 (1997).
  * Baez (2021) J. C. Baez, Entropy 23, 1436 (2021).
  * Frank and Slatkin (1992) S. A. Frank and M. Slatkin, Trends in Ecology & Evolution 7, 92 (1992).
  * Price (1972) G. R. Price, Annals of human genetics 35, 485 (1972).
  * Grafen (2000) A. Grafen, Proceedings of the Royal Society of London. Series B: Biological Sciences 267, 1223 (2000).
  * Frank (2012a) S. A. Frank, Journal of Evolutionary Biology 25, 1002 (2012a).
  * Baum (2017) W. M. Baum, Journal of the Experimental Analysis of Behavior 107, 321 (2017).
  * Frank (2018) S. A. Frank, Entropy 20, 978 (2018).
  * Frank and Bruggeman (2020) S. A. Frank and F. J. Bruggeman, Entropy 22, 1395 (2020).
  * Malthus (1872) T. R. Malthus,  _An Essay on the Principle of Population.._ (1872).
  * Leibler and Kussell (2010) S. Leibler and E. Kussell, Proceedings of the National Academy of Sciences 107, 13183 (2010).
  * Crow and Kimura (2017) J. F. Crow and M. Kimura,  _An introduction to population genetics theory, 1970_ (Scientific Publishers, 2017).
  * Basener and Sanford (2018) W. F. Basener and J. C. Sanford, Journal of Mathematical Biology 76, 1589 (2018).
  * Sekimoto (2010) K. Sekimoto,  _Stochastic energetics_ , Vol. 799 (Springer, 2010).
  * Seifert (2012) U. Seifert, Reports on progress in physics 75, 126001 (2012).
  * Strogatz (2018) S. H. Strogatz,  _Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering_ (CRC press, 2018).
  * Wakamoto  _et al._ (2012) Y. Wakamoto, A. Y. Grosberg, and E. Kussell, Evolution: International Journal of Organic Evolution 66, 115 (2012).
  * Lambert and Kussell (2015) G. Lambert and E. Kussell, Physical review X 5, 011016 (2015).
  * Hoffmann and Olivier (2016) M. Hoffmann and A. Olivier, Stochastic Processes and their Applications 126, 1433 (2016).
  * Nozoe  _et al._ (2017) T. Nozoe, E. Kussell, and Y. Wakamoto, PLoS genetics 13, e1006653 (2017).
  * Andrae  _et al._ (2010) B. Andrae, J. Cremer, T. Reichenbach, and E. Frey, Physical review letters 104, 218102 (2010).
  * Qian (2014) H. Qian, Quantitative Biology 2, 47 (2014).
  * Kobayashi and Sughiyama (2015) T. J. Kobayashi and Y. Sughiyama, Physical review letters 115, 238102 (2015).
  * Sughiyama  _et al._ (2015) Y. Sughiyama, T. J. Kobayashi, K. Tsumura, and K. Aihara, Physical Review E 91, 032120 (2015).
  * Sughiyama and Kobayashi (2017) Y. Sughiyama and T. J. Kobayashi, Physical Review E 95, 012131 (2017).
  * Kobayashi and Sughiyama (2017) T. J. Kobayashi and Y. Sughiyama, Physical Review E 96, 012402 (2017).
  * García-García  _et al._ (2019) R. García-García, A. Genthon, and D. Lacoste, Physical Review E 99, 042413 (2019).
  * Genthon and Lacoste (2020) A. Genthon and D. Lacoste, Scientific Reports 10, 1 (2020).
  * Genthon and Lacoste (2021) A. Genthon and D. Lacoste, Physical Review Research 3, 023187 (2021).
  * Yoshimura and Ito (2021a) K. Yoshimura and S. Ito, Physical Review Letters 127, 160601 (2021a).
  * Kolchinsky (2021) A. Kolchinsky, arXiv preprint arXiv:2112.02809 (2021).
  * Esposito and Van den Broeck (2010) M. Esposito and C. Van den Broeck, Physical Review E 82, 011143 (2010).
  * Van den Broeck and Esposito (2010) C. Van den Broeck and M. Esposito, Physical Review E 82, 011144 (2010).
  * Barato and Seifert (2015) A. C. Barato and U. Seifert, Physical review letters 114, 158101 (2015).
  * Zhang  _et al._ (2020) Z. Zhang, S. Guan, and H. Shi, Journal of Statistical Mechanics: Theory and Experiment 2020, 073501 (2020).
  * Crooks (2007) G. E. Crooks, Physical Review Letters 99, 100602 (2007).
  * Ito (2018) S. Ito, Physical review letters 121, 030605 (2018).
  * Mandelstam and Tamm (1991) L. Mandelstam and I. Tamm, in  _Selected papers_ (Springer, 1991) pp. 115–123.
  * Anandan and Aharonov (1990) J. Anandan and Y. Aharonov, Physical review letters 65, 1697 (1990).
  * Margolus and Levitin (1998) N. Margolus and L. B. Levitin, Physica D: Nonlinear Phenomena 120, 188 (1998).
  * Taddei  _et al._ (2013) M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Physical review letters 110, 050402 (2013).
  * Pires  _et al._ (2016) D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, Physical Review X 6, 021031 (2016).
  * Shanahan  _et al._ (2018) B. Shanahan, A. Chenu, N. Margolus, and A. Del Campo, Physical review letters 120, 070401 (2018).
  * Okuyama and Ohzeki (2018) M. Okuyama and M. Ohzeki, Physical review letters 120, 070402 (2018).
  * Amari (2016) S.-i. Amari,  _Information geometry and its applications_ , Vol. 194 (Springer, 2016).
  * Ito and Dechant (2020) S. Ito and A. Dechant, Physical Review X 10, 021056 (2020).
  * Nicholson  _et al._ (2020) S. B. Nicholson, L. P. Garcia-Pintos, A. del Campo, and J. R. Green, Nature Physics 16, 1211 (2020).
  * Yoshimura and Ito (2021b) K. Yoshimura and S. Ito, Physical Review Research 3, 013175 (2021b).
  * Ito (2022) S. Ito, Journal of Physics A: Mathematical and Theoretical 55, 054001 (2022).
  * Ashida  _et al._ (2021) K. Ashida, K. Aoki, and S. Ito, bioRxiv , 2020 (2021).
  * Rao (1992) C. R. Rao, in  _Breakthroughs in statistics_ (Springer, 1992) pp. 235–247.
  * Sato  _et al._ (2003) K. Sato, Y. Ito, T. Yomo, and K. Kaneko, Proceedings of the National Academy of Sciences 100, 14086 (2003).
  * Michel  _et al._ (2005) P. Michel, S. Mischler, and B. Perthame, Journal de mathématiques pures et appliquées 84, 1235 (2005).
  * Wolf  _et al._ (2005) D. M. Wolf, V. V. Vazirani, and A. P. Arkin, Journal of theoretical biology 234, 227 (2005).
  * Kussell and Leibler (2005) E. Kussell and S. Leibler, Science 309, 2075 (2005).
  * Rivoire and Leibler (2011) O. Rivoire and S. Leibler, Journal of Statistical Physics 142, 1124 (2011).
  * Frank (2012b) S. A. Frank, Journal of Evolutionary Biology 25, 2377 (2012b).
  * Frank (2013) S. A. Frank, Journal of Evolutionary Biology 26, 457 (2013).
  * Kaneko  _et al._ (2015) K. Kaneko, C. Furusawa, and T. Yomo, Physical Review X 5, 011014 (2015).
  * Xue and Leibler (2018) B. Xue and S. Leibler, Proceedings of the National Academy of Sciences 115, 12745 (2018).
  * Nakashima and Kobayashi (2022) S. Nakashima and T. J. Kobayashi, Physical Review Research 4, 013069 (2022).
  * Adachi  _et al._ (2022) K. Adachi, R. Iritani, and R. Hamazaki, Communications Physics 5, 1 (2022).
  * García-Pintos (2022) L. P. García-Pintos, arXiv preprint arXiv:2202.07533 (2022).
  * Note (1) We remark that the Cramér–Rao bound holds even for time-dependent observables by changing the definition of $v_{R}$ as $v_{R}\,{=}\,(d_{t}\langle R\rangle-\langle d_{t}R\rangle)/\sqrt{\mathrm{Var}[R]}$.
  * Cover (1999) T. M. Cover,  _Elements of information theory_ (John Wiley & Sons, 1999).
  * Frank (2009) S. A. Frank, Journal of Evolutionary Biology 22, 231 (2009).
  * Note (2) We remark that a similar concept of the angle can be seen in the Price equation Frank (2012a).
  * (85) _See Supplemental Material athttp://link.aps.org/supplemental/... for more detailed discussions on the derivation of Eq. (16), the mutation bound, the precise evaluation of the speed limits, and how to test our results experimentally._
  * Wiseman (1994) H. M. Wiseman,  _Quantum trajectories and feedback_ , Ph.D. thesis, University of Queensland (1994).
  * Cresser  _et al._ (2006) J. D. Cresser, S. M. Barnett, J. Jeffers, and D. T. Pegg, Optics Communications 264, 352 (2006).
  * Zhou (2022) Y.-N. Zhou, arXiv preprint arXiv:2204.09049 (2022).
  * Baake and Georgii (2007) E. Baake and H.-O. Georgii, Journal of mathematical biology 54, 257 (2007).

Supplemental Material

##  Appendix A Evaluation of the speed limits

Here, we explain the derivation of Eq. (16). Taking the difference between
$v_{\rm info}$ and $v_{R}^{\lambda}+v_{\rm info}^{\mathsf{W}}$, we get $v_{\rm
info}-\quantity(v_{R}^{\lambda}+v_{\rm info}^{\mathsf{W}})=v_{\rm info}-v_{\rm
info}^{\mathsf{W}}-\cos\theta_{R}^{\lambda}v_{\rm info}^{\lambda}$. Thus, the
following equations hold.

$\displaystyle\cos\theta_{R}^{\lambda}\leq\dfrac{v_{\rm info}-v_{\rm
info}^{\mathsf{W}}}{v_{\rm info}^{\lambda}}$
$\displaystyle\implies(v_{R}\leq\;)\;v_{R}^{\lambda}+v_{\rm
info}^{\mathsf{W}}\leq v_{\rm info},$
$\displaystyle\cos\theta_{R}^{\lambda}\geq\dfrac{v_{\rm info}-v_{\rm
info}^{\mathsf{W}}}{v_{\rm info}^{\lambda}}$
$\displaystyle\implies(v_{R}\leq\;)\;v_{\rm info}\leq v_{R}^{\lambda}+v_{\rm
info}^{\mathsf{W}}.$ (18)

The same calculations can be applied to the lower limits: $-v_{\rm info}$ and
$v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}}$. The difference between the two
is, $\quantity(v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}})-(-v_{\rm
info})=v_{\rm info}+\cos\theta_{R}^{\lambda}v_{\rm info}^{\lambda}-v_{\rm
info}^{\mathsf{W}}$. Therefore,

$\displaystyle\cos\theta_{R}^{\lambda}\geq-\dfrac{v_{\rm info}-v_{\rm
info}^{\mathsf{W}}}{v_{\rm info}^{\lambda}}$ $\displaystyle\implies-v_{\rm
info}\leq v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}}\;(\;\leq v_{R}),$
$\displaystyle\cos\theta_{R}^{\lambda}\leq-\dfrac{v_{\rm info}-v_{\rm
info}^{\mathsf{W}}}{v_{\rm info}^{\lambda}}$ $\displaystyle\implies
v_{R}^{\lambda}-v_{\rm info}^{\mathsf{W}}\leq-v_{\rm info}\;(\;\leq v_{R}).$
(19)

Taking the $\arccos$ on both sides of these equations and using the relation
$\arccos(-x)=\pi-\arccos(x)$, we obtain Eq. (16) in the main text.

##  Appendix B Detail of the mutation bound

In this section, we describe the results of the numerical calculation for the
following speed limit, which we call the mutation bound:

$\displaystyle v_{R}^{\mathsf{W}}-v_{\rm info}^{\lambda}\leq v_{R}\leq
v_{R}^{\mathsf{W}}+v_{\rm info}^{\lambda}.$ (20)

This speed limit is expected to give a good evaluation in mutation-dominant
situations, whereas the selection bound gives a good evaluation in selection-
dominant situations. We below proceed with the discussion in parallel with
that in the main text. We define the angle $\theta_{R}^{\mathsf{W}}$ between
observable $R$ and the stochastic entropy change rate in the system due to
mutation $-\dot{\sigma}^{\mathsf{W}}$ as

$\displaystyle\theta_{R}^{\mathsf{W}}=\arccos\left(\frac{\langle\Delta
R,-\dot{\sigma}^{\mathsf{W}}\rangle}{\|\Delta
R\|\|-\dot{\sigma}^{\mathsf{W}}\|}\right)=\arccos\left(\frac{v_{R}^{\mathsf{W}}}{v_{\rm
info}^{\mathsf{W}}}\right).$ (21)

Using this angle $\theta_{R}^{\mathsf{W}}$ and another angle
$\theta^{\dagger}$ defined as

$\displaystyle\theta^{{\dagger}}\coloneqq\arccos\quantity(\dfrac{v_{\rm
info}-v_{\rm info}^{\lambda}}{v_{\rm info}^{\mathsf{W}}}),$ (22)

the relations among the upper bound $v_{\rm
info}^{\lambda}+v_{R}^{\mathsf{W}}$ and the lower bound $-v_{\rm
info}^{\lambda}+v_{R}^{\mathsf{W}}$ by the mutation bound and $v_{\rm info}$
are expressed as

$\displaystyle\begin{cases}0\leq\theta_{R}^{\mathsf{W}}\leq\theta^{\dagger}&\implies
v_{R}\leq v_{\rm info}\leq v_{R}^{\mathsf{W}}+v_{\rm info}^{\lambda}\\\
\theta^{\dagger}\leq\theta_{R}^{\lambda}\leq\pi&\implies v_{R}\leq
v_{R}^{\mathsf{W}}+v_{\rm info}^{\lambda}\leq v_{\rm info}\\\
0\leq\theta_{R}^{\lambda}\leq\pi-\theta^{\dagger}&\implies-v_{\rm info}\leq
v_{R}^{\mathsf{W}}-v_{\rm info}^{\lambda}\leq v_{R}\\\
\pi-\theta^{\dagger}\leq\theta_{R}^{\lambda}\leq\pi&\implies
v_{R}^{\mathsf{W}}-v_{\rm info}^{\lambda}\leq-v_{\rm info}\leq
v_{R}\end{cases}.$ (23)

In contrast to $\theta^{\ast}$, $\theta^{\dagger}$ becomes small when the
contribution of mutation to the evolution of the probability distribution is
large. It indicates that the range of $\theta_{R}^{\mathsf{W}}$ where the
mutation bound is tighter, $[\theta^{\dagger},\pi-\theta^{\dagger}]$, gets
wider. To sum up, the selection bound gets tighter when the contribution of
selection is large, while the mutation bound gets tighter when the
contribution of mutation is large.

With the parameters used to demonstrate the selection bound in the main text,
the mutation bound gives a loose bound. As shown in Figs. 5 and 5, the
mutation bound is loose in situations where the selection bound gives a tight
evaluation, and conversely, the selection bound is loose in situations where
the mutation bound is tight.

Figure 4: Numerical calculation of a situation where the selection bound can
be tighter for $v_{R}$ than Cramér–Rao bound and the mutation bound cannot.
The horizontal axis is time $t$ and each value is plotted in the upper part of
the graph. The lower part shows the angles. The number of species are $n=10$,
the observable $R$, growth rate $\lambda$, and the off-diagonal components of
the transition matrix $\mathsf{W}$ are taken as uniform random numbers in
$[-10,10],[-10,10],\mathrm{and}[0.0.1]$, respectively. The initial
distribution is also generated as uniform random numbers.

Figure 5: Numerical calculation of a situation where the mutation bound can be
tighter for $v_{R}$ than Cramér–Rao bound and the selection bound cannot. The
horizontal axis is time $t$ and each value is plotted in the upper part of the
graph. The lower part shows the angles. Compared to Fig. 5, the parameters are
different. The observable $R$, growth rate $\lambda$, transition rate
$\mathsf{W}$ is generated as uniform random numbers in
$[-10,10],[-5,5],\mathrm{and}[0,5]$, respectively. Note that the ratio of the
growth rate to the mutation rate is $1$, not $100$ as in Fig. 3 in the main
text and Fig. 5.

Thus, the selection bound and the mutation bound provide good bounds to
evaluate the change speed of the observables in different situations.

##  Appendix C How the strength of selection and mutation affect the
evaluation of the speed limits

In order to evaluate the tightness of the selection bound and the mutation
bound quantitatively, we discuss the dependence of the “tendency” to give
better limits on the ratio of $\lambda_{\rm max}$ to $\mathsf{W}_{\rm max}$.
This tendency for the selection bound to be tighter than the Cramér–Rao bound
can be measured using the value of $\theta^{\ast}$. The selection bound gets
tighter when the angle $\theta_{R}^{\lambda}$ is in the range
$[\theta^{\ast},\pi-\theta^{\ast}]$. Therefore, if we define
$\mathcal{P}^{\ast}\coloneqq\max\\{(\pi-2\theta^{\ast})/\pi,0\\}$, this
$\mathcal{P}^{\ast}$ quantifies the tendency of the selection bound to give a
better evaluation. The selection bound tends to be tighter when
$\mathcal{P}^{\ast}$ is close to $1$ and looser when $\mathcal{P}^{\ast}$ is
close to $0$. Note that $\theta^{\ast}$ is not always under $\pi/2$, thus,
$\pi-2\theta^{\ast}$ could be negative. In the same way, the tendency of the
mutation bound to give a better evaluation can be measured by the value
defined as
$\mathcal{P}^{\dagger}\coloneqq\max\\{(\pi-2\theta^{\dagger})/\pi,0\\}$. It is
noteworthy that $\mathcal{P}^{\ast}$ and $\mathcal{P}^{\dagger}$ do not depend
on the observable $R$.

We demonstrate the dependence of $\mathcal{P}^{\ast}$ and
$\mathcal{P}^{\dagger}$ on the ratio of $\lambda_{\rm max}$ to
$\mathsf{W}_{\rm max}$ (see Fig. 6). In the numerical calculation, $\lambda$
and $\mathsf{W}$ is generated by uniform random values in the range
$[-\lambda_{\rm max},\lambda_{\rm max}]$, and $[0,\mathsf{W}_{\rm max}]$,
respectively. Note here that both $\mathcal{P}^{\ast}$ and
$\mathcal{P}^{\dagger}$ depend on time, so we use their values in the initial
state of the dynamics. Fig. 6 shows that the tendency $\mathcal{P}^{\ast}$ of
the selection bound to get tighter than the Cramér–Rao bound increases when
$\lambda_{\rm max}$ is larger than $\mathsf{W}_{\rm max}$. This calculation
confirms the statement in the main text that the selection bound is effective
when natural selection is dominant.

Another finding is that the point where the two curves of $\mathcal{P}^{\ast}$
and $\mathcal{P}^{\dagger}$ intersect is governed by the number of species
$n$. This point represents the ratio $\lambda_{\rm max}/\mathsf{W}_{\rm max}$
with $v_{\rm info}^{\lambda}=v_{\rm info}^{\mathsf{W}}$. Moreover, the
tendency for the selection/mutation bound to give better evaluation swaps at
this point. Fig. 6 shows that such a ratio is not $1$, but close to the number
of species $n$. It implies that the contribution of selection and that of
mutation compete when $\lambda_{\rm max}$ is about $n$ times larger than
$\mathsf{W}_{\rm max}$. This fact can be understood by considering a situation
where the parameters are $\lambda_{i}=1,\;W_{ij}=1(i\neq j)$. The contribution
to the growth rate of trait $i$ of the selection is $1$, whereas the
contribution of the mutation is $n-1$. Therefore, if the traits grow at an
identical rate, the contribution of selection becomes larger in small
communities.

Figure 6: Numerical calculation for $\mathcal{P}^{\ast}$ and
$\mathcal{P}^{\dagger}$, that quantify the tendency of the selection/mutation
bound to get tighter than the Cramér–Rao bound. $\lambda$ and $\mathsf{W}$ are
generated by uniform random numbers in $[-\lambda_{\rm max},\lambda_{\rm
max}]$ and $[0,\mathsf{W}_{\rm max}]$, respectively. The graphs show how
$\mathcal{P}^{\ast}$ and $\mathcal{P}^{\dagger}$ depend on the ratio of
$\lambda_{\rm max}$ to $\mathsf{W}_{\rm max}$. $\mathcal{P}_{\rm lim}$ is the
limit of $\mathcal{P}^{\ast}$ as $\lambda_{\rm max}/\mathsf{W}_{\rm max}\to 0$
or that of $\mathcal{P}^{\dagger}$ as $\lambda_{\rm max}/\mathsf{W}_{\rm
max}\to\infty$. These two values converge to an identical value, which does
not necessarily go to $0$. Each graph is the result for a different number of
species $n=10,100,1000$.

##  Appendix D How to verify the selection bound and the mutation bound from
experimental data

In this section, we describe how one can quantitatively test our speed limit
using single-cell genealogical data in the form of population lineage trees
which include data of cell divisions and mutations or phenotypic switching
(see Fig. 7 as an example). We can compute the number $N_{i}(K,t;t^{\prime})$
of individuals with $i$-th trait at time $t$ that have experienced $K$
divisions since time $t^{\prime}$ from single-cell genealogical data.
$N_{i}(K,t;t^{\prime})$ enable us to compute the instantaneous values of all
the quantities in our speed limit, i.e., $p_{i}(t)$,
${\dot{\sigma}}_{i}^{\lambda}$, ${\dot{\sigma}}_{i}^{\mathsf{W}}$, and
${\dot{\sigma}}_{i}^{\lambda}$ as well as
$v_{R}^{\lambda},v_{R}^{\mathsf{W}},v_{\rm info}^{\lambda}$ and $v_{\rm
info}^{\mathsf{W}}$ for an arbitrary observable $R$. The details are as
follows.

Firstly, from $N_{i}(K,t;t^{\prime})$, we define chronological(forward)
distribution $p^{\rm ch}$ as

$\displaystyle p_{i}^{\rm
ch}(t;t^{\prime})\coloneqq\sum_{K=0}^{\infty}\frac{N_{i}(K,t;t^{\prime})}{N_{\rm
tot}(t^{\prime})2^{K}}.$ (24)

From these values, we can obtain the instantaneous values of $N_{i}(t)$ and
$p_{i}(t)$ for each subpopulation: the sum of $N_{i}(K,t;t^{\prime})$ with
respect to $K$ equals $N_{i}(t)$,

$\displaystyle\sum_{K=0}^{\infty}N_{i}(K,t;t^{\prime})=N_{i}(t),$ (25)

and $p_{i}^{\rm ch}(t;t^{\prime})$ is equal to $p_{i}(t)$ when $t^{\prime}=t$,

$\displaystyle p_{i}^{\rm ch}(t;t)=p_{i}(t),$ (26)

because only $K=0$ is allowed as the number of division between $t$ and $t$.
Note that the chronological distribution defined above has been studied not
only as a theoretical object Baake and Georgii (2007); Kobayashi and Sughiyama
(2015) but also utilized to analyze experimental data as in Refs. Nozoe  _et
al._ (2017); Genthon and Lacoste (2020). $t^{\prime}$ is usually set to the
initial time and not explicitly written in these previous studies.

Up to here, we can compute speeds $v_{R}^{\lambda}$, $v_{R}^{\mathsf{W}}$,
$v_{\mathrm{info}}^{\lambda}$ and $v_{\mathrm{info}}^{\mathsf{W}}$ for any
observable $R$ by using $p_{i}(t)$. On the other hand, since the remaining
quantities, $\dot{\sigma}_{i}$, $\dot{\sigma}_{i}^{\lambda}$, and
$\dot{\sigma}_{i}^{\mathsf{W}}$, depend on the other parameters, $\lambda$ and
$\mathsf{W}$, we need more calculations. Surprisingly, single-cell
genealogical data enable us to compute them without estimating the parameters,
as we show below.

Figure 7:  Example of single-cell lineage trees with two traits, $0$ (green)
and $1$ (gray). The distributions $p(t)$ and $p^{\rm ch}(t)$ are available
only from the cell lineage by counting $N_{i}(K,t;t^{\prime})$. The blue lines
show the trajectories of individuals that experienced $1$ divisions between
time $t^{\prime}$ and $t$ in the subpopulation with the trait $0$ at time $t$.
Thus $N_{0}(1,t;t^{\prime})$ is $2$ in this cell lineage.

As a preparation, let us consider the differential equations that
$N_{i}(K,t;t^{\prime})$ and $p^{\mathrm{ch}}$ satisfy. Let an individual with
$i$th trait divide into $2$ individuals at rate $r_{i}\geq 0$. Then, by
division, $N_{i}(K,t;t^{\prime})$ increases by
$r_{i}N_{i}(K-1,t;t^{\prime})dt$ in an infinitesimal duration $dt$, while
decreasing by $r_{i}N_{i}(K,t;t^{\prime})dt$. Note that only one division can
occur in an infinitesimal duration $dt$. By taking into account the term due
to mutation $\sum_{j=1}^{n}W_{ij}N_{j}(K,t;t^{\prime})$, we get the equation

$\displaystyle\frac{d}{dt}N_{i}(K,t;t^{\prime})=2r_{i}N_{i}(K-1,t;t^{\prime})-r_{i}N_{i}(K,t;t^{\prime})+\sum_{j=1}^{n}W_{ij}N_{j}(K,t;t^{\prime}),$
(27)

where we define $N_{i}(K,t;t^{\prime})=0$ for $K<0$. By taking the sum for
$K$, we obtain

$\displaystyle\frac{d}{dt}N_{i}(t)=r_{i}N_{i}(t)+\sum_{j=1}^{n}W_{ij}N_{j}(t).$
(28)

If we write $r_{i}$ as $\lambda_{i}$, it is identical to our model in the main
text. Note that $r_{i}$ is all non-negative, while $\lambda_{i}$ can be
negative values in general, which reflects the experimental setup that does
not account for individual mortality. On the other hand, combining Eq. (27)
with Eq. (24), we find

$\displaystyle\frac{d}{dt}p^{\rm ch}_{i}(t;t^{\prime})$
$\displaystyle=\sum_{K=0}^{\infty}\frac{1}{N_{\rm
tot}(t^{\prime})2^{K}}\quantity[2r_{i}N_{i}(K-1,t;t^{\prime})-r_{i}N_{i}(K,t;t^{\prime})+\sum_{j=1}^{n}W_{ij}N_{j}(K,t;t^{\prime})]$
$\displaystyle=r_{i}\sum_{K=0}^{\infty}\frac{N_{i}(K-k_{i},t;t^{\prime})}{N_{\rm
tot}(t^{\prime})2^{K-1}}-r_{i}\sum_{K=0}^{\infty}\frac{N_{i}(K,t;t^{\prime})}{N_{\rm
tot}(t^{\prime})2^{K}}+\sum_{j=1}^{n}W_{ij}\sum_{K=0}^{\infty}\frac{N_{j}(K,t;t^{\prime})}{N_{\rm
tot}(t^{\prime})2^{K}}$ $\displaystyle=\sum_{j=1}^{n}W_{ij}p^{\rm
ch}_{j}(t;t^{\prime}).$ (29)

Now we can present the way to compute ${\dot{\sigma}}_{i}^{\mathsf{W}}$ and
${\dot{\sigma}}_{i}^{\lambda}$, using only the distributions $p_{i}(t)$ and
$p_{i}^{\rm ch}(t;t^{\prime})$, which are available from single-cell lineage
tree data. Discretizing Eq. (29) and dividing the obtained equation by
$p_{i}(t)$, we find the relation

$\displaystyle\frac{p_{i}^{\rm ch}(t+dt;t^{\prime})-p_{i}^{\rm
ch}(t;t^{\prime})}{p_{i}(t)dt}=\sum_{j=1}^{n}W_{ij}\frac{p^{\rm
ch}_{j}(t;t^{\prime})}{p_{i}(t)}.$ (30)

Then, substituting $t$ for $t^{\prime}$ and using Eq. (26), we see that we can
calculate ${\dot{\sigma}}_{i}^{\mathsf{W}}$ as

$\displaystyle\frac{p_{i}^{\rm
ch}(t+dt;t)-p_{i}(t)}{p_{i}(t)dt}=\sum_{j=1}^{n}W_{ij}\frac{p_{j}(t)}{p_{i}(t)}=-{\dot{\sigma}}_{i}^{\mathsf{W}}.$
(31)

On the other hand, if we consider the discretization of Eq. (28), as we did
for Eq. (29), we obtain ${\dot{\sigma}}_{i}$ as

$\displaystyle\frac{p_{i}(t+dt)-p_{i}(t)}{p_{i}(t)dt}=\Delta\lambda_{i}+\sum_{j=1}^{n}W_{ij}\frac{p_{j}(t)}{p_{i}(t)}=-{\dot{\sigma}}_{i}.$
(32)

Then ${\dot{\sigma}}_{i}^{\lambda}$ is given by
${\dot{\sigma}}_{i}^{\lambda}={\dot{\sigma}}_{i}-{\dot{\sigma}}_{i}^{\mathsf{W}}$.
As a result, it is finally shown that we can compute all the relevant
quantities we discuss in the main text from single-cell genealogical data.