On Eigen’s Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces

Abstract

A two-valued fitness landscape is introduced for the classical Eigen’s quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is $2^N\times 2^N$, where N is the sequence length. It is shown that if the fitness function is equal to $w+s$ on a G-orbit A and is equal to w elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most $N+1$. Here G is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length N with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the G-orbit A. Motivated by the analysis of the two-valued fitness landscapes, an abstract generalization of Eigen’s model is introduced such that the sequences are identified with the points of a finite metric space X together with a group of isometries acting transitively on X. In particular, a simplicial analog of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.

Appendix: The Infinite Sequence Limit \(N\rightarrow \infty \)

In this appendix, we present the discussion of the error threshold (Hermisson et al. 2002; Wilke 2005) of the classical Eigen’s quasispecies model in terms of the sequences of orbits. While this discussion is rather technical, it, in our view, clarifies the geometric picture of the error threshold and allows direct generalization for other mutational landscapes, including the simplicial one, presented in Sect. 6.3.3. We remark at the beginning of this section that all the presented results are proved for the introduced two-valued fitness landscapes where the higher fitnesses correspond to the sequences in a given G-orbit. For more general situations, these results are likely to change quantitatively; one example of such more general situation is given in Sect. 2, where the fitness landscapes is defined on more than two orbits (under preparation).

In Corollary 4.3, we obtained the algebraic equation (26) of degree at most $N+1$ for the leading eigenvalue $\overline{w}=\overline{w}(q)$. The advantage of having a polynomial equation of degree $N+1$ notwithstanding, solving (26) becomes complicated as $N\rightarrow \infty$. Moreover, it is well known that at least for some fitness landscapes (including the classical single-peaked fitness landscape) the phenomenon of the error threshold is observed: There exists a critical mutation rate q, after which the quasispecies distribution ${\varvec{p}}$ becomes uniform. This phenomenon is usually identified with a non-analytical behavior of the limiting eigenvalue $\overline{w}$ when $N\rightarrow \infty$, a general idea can be grasped from Fig. 1b, where it is seen that there exists a corner point on the graph of the function $\overline{w}$.

In this section, we propose several steps to rigorously define and analyze this kind of behavior in terms of sequences of orbits $A_n$ that determine our two-valued fitness landscapes. First, we find some bounds for the function $\overline{w}$ provided $0.5\le q\le 1$. Next, we restrict our attention at the special class of sequences $(A_n)_{n=n_0}^\infty$, which we call admissible and of the moderate growth (here $n_0$ is a sufficiently large natural number). Finally, among all those admissible sequences of the moderate growth we identify the ones that demonstrate some kind of non-uniform convergence for the corresponding sequence of eigenvalues $(\overline{w}^{(n)})_{n=n_0}^\infty$.

1.1 Lower and Upper Bounds on $\overline{w}(q)$

First, we note that for our purposes it is easier to deal with the series (25) rather than (26). We also make the following substitutions

$$\begin{aligned} w=us\;,\quad \overline{w}=\overline{u}s. \end{aligned}$$

(81)

Then (25) turns into

$$\begin{aligned} \overline{u}= \sum _{c=0}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c F_A\left( (2q-1)^{c+1}\right) , \end{aligned}$$

(82)

where the polynomial $F_A(z)$, defined in (21), can be represented in the form (24).

From Example 5.9, we have that $sF_A(2q-1)=\overline{w}(q)$ is the leading eigenvalue if $w=0$. It was proved in Semenov et al. (2014) that $\overline{w}(q)$ increases on the segment $0.5\le q\le 1$. Therefore, on this segment we have the non-increasing sequence (for any fixed q)

$$\begin{aligned} F_A(2q-1)\ge F_A\left( (2q-1)^2\right) \ge \dots \ge F_A\left( (2q-1)^{c})\ge F_A((2q-1)^{c+1}\right) \ge \dots >0, \end{aligned}$$

(83)

since $F_A((2q-1)^{c})>0$ according to (22). Hence,

$$\begin{aligned} \overline{u}= \sum _{c=0}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c F_A((2q-1)^{c+1})\le F_A(2q-1)\sum _{c=0}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c= \frac{F_A((2q-1))\,\overline{u}}{\overline{u}-u}. \end{aligned}$$

(84)

It follows that $\overline{u}\le u+ F_A(2q-1)$, or

$$\begin{aligned} \overline{w}(q)\le w+sF_A(2q-1)=:\overline{w}_{up,1}(q). \end{aligned}$$

A second upper bound can be obtained as follows:

$$\begin{aligned} \overline{u}&= \sum _{c=0}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c F_A\left( (2q-1)^{c+1})=F_A(2q-1)+ \sum _{c=1}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c F_A((2q-1)^{c+1}\right) \\&\le F_A(2q-1)+F_A\left( (2q-1)^2\right) \sum _{c=1}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c= F_A(2q-1)+\frac{uF_A((2q-1)^2)}{\overline{u}-u}. \end{aligned}$$

Solving the quadratic inequality, we get

$$\begin{aligned} \overline{u}\le \frac{u+F_A(2q-1)+\sqrt{(u+F_A(2q-1))^2-4u(F_A(2q-1)-F_A((2q-1)^2))}}{2}, \end{aligned}$$

or,

$$\begin{aligned} \overline{w}(q)\le \frac{\overline{w}_{up,1}(q)+ \sqrt{\overline{w}^2_{up,1}(q)-4w(sF_A(2q-1)-sF_A((2q-1)^2))}}{2}=:\overline{w}_{up,2}(q). \end{aligned}$$

(85)

Remark 7.1

In view of (83) $\overline{w}_{up,2}(q)\le \overline{w}_{up,1}(q)$.

To obtain a lower bound on $\overline{w}(q)$, we use the approach applied in Semenov et al. (2014). Since $\overline{w}(q)$ increases on the segment $0.5\le q\le 1$, therefore

$$\begin{aligned} \overline{w}(q)\ge \overline{w}(0.5)=w+\frac{s|A|}{2^N}. \end{aligned}$$

(86)

By the definition of (21)

$$\begin{aligned} F_A((2q-1)^{c+1})&=\sum _{b\in A}\left( \frac{1-(2q-1)^{c+1}}{2}\right) ^{H_{ab}} \left( \frac{1+(2q-1)^{c+1}}{2}\right) ^{N-H_{ab}}\\&\ge \left( \frac{1+(2q-1)^{c+1}}{2}\right) ^{N}\ge \left( \frac{1+(2q-1)}{2}\right) ^{(c+1)N}=q^{(c+1)N}, \end{aligned}$$

since $a\in A$, $H_{aa}=0$ and the function $f(t)=t^{c+1}$ is convex (downward) on the segment [0, 1].

Now from (82)

$$\begin{aligned} \overline{u}\ge \sum _{c=0}^{\infty }\left( \frac{u}{\overline{u}}\right) ^c q^{(c+1)N}=\frac{\overline{u}\,q^N}{\overline{u}-uq^N},\quad \text{ or }\;\; \overline{u}\ge (u+1)q^N,\quad \text{ or }\;\; \overline{w}\ge (w+s)q^N. \end{aligned}$$

(87)

Combining (86) and (87) yields

$$\begin{aligned} \overline{w}(q)\ge \max \left( w+\frac{s|A|}{2^N}, (w+s)\,q^N\right) =:\overline{w}_{low}(q). \end{aligned}$$

(88)

Thus we have proved

Proposition 7.2

For the leading eigenvalue $\overline{w}(q)$ of (3) in the case of the two-valued fitness landscape, we have

$$\begin{aligned} \overline{w}_{low}(q)\le w(q)\le \overline{w}_{up,2}(q),\quad 0.5\le q\le 1, \end{aligned}$$

where $\overline{w}_{low}(q)$ is given by (88) and $\overline{w}_{up,2}(q)$ is given by (85).

A numerical example with the obtained bounds is given in Fig. 4.

Fig. 4

(Color figure online) The lower and upper bounds on the leading eigenvalue $\overline{w}$ in the case of the quaternion landscape (Example 5.8), (a) $N=8$, (b) $N=50$

1.2 Admissible Sequences of Orbits

To make a progress in analyzing the limit behavior of our system, when $N\rightarrow \infty$ we introduce in this subsection two definitions in terms of which this behavior will be described.

From the previous subsection, we see that the curve $\overline{w}=\overline{w}_{low}(q)$ has a corner point on [0.5, 1], which we denote $q_*$:

$$\begin{aligned} q_*=q_*^{(N)}=\root N \of {\frac{w+s|A|2^{-N}}{w+s}}=\root N \of {\frac{u+|A|2^{-N}}{u+1}}=\root N \of {\frac{\overline{w}(0.5)}{\overline{w}(1)}}. \end{aligned}$$

(89)

The function $\overline{w}_{low}(q)$ is constant for $0.5\le q\le q_*$ and increases for $q_*<q\le 1$ (see Fig. 4). It was shown in Semenov et al. (2014) that for the single-peaked landscapes ($|A|=1$) the lower bound $\overline{w}_{low}(q)$ provides a close approximation for $\overline{w}(q)$ for sufficiently large N. Our goal is to generalize these results on the case of the two-valued fitness landscapes.

From this point on, we shall use n as the index, which tends to infinity. In most cases, it actually coincides with the sequence length N, albeit not always, hence the choice of notation.

One of the main underlying questions concerning the quasispecies model and especially its infinite sequence limit, is how actually the fitness landscape is scaled when $N\rightarrow \infty$. In most works in the literature a continuous limit is used, which basically narrows the pull of the allowed fitness landscapes to the ones which have, given this continuous limit, a limit fitness function, which must be also continuous [e.g., Baake and Georgii (2007) and Saakian and Hu (2006)]. Here we take a different approach by specifying sequences of orbits $(A_n)_{n=n_0}^\infty$, on which the fitness landscape is defined. The sequences that are of interest to us will be called admissible.

Suppose that for any $n\ge n_0$ a sequence of $G_n$-orbits $A_n\in X_n$ is given, where $G_n\leqslant \mathrm{Iso}\, (X_n)$. When $n\rightarrow \infty$ the group $\mathrm{Iso}\, (X_n)$ will be always viewed as a subgroup of $\mathrm{Iso}\, (X_{n+1})$. More precisely, let $g\in \mathrm{Iso}\, (X_n)$ be a fixed isometry and let $a\in X_{n+1}$ be represented as $a=a_{n}+\alpha _{n} 2^{n}$ where $a_{n},\alpha _{n}\in X_n$. Then g, viewed as an element of $\mathrm{Iso}\, (X_{n+1})$, maps $a\in X_{n+1}$ to $g(a):=g(a_{n})+g(\alpha _{n}) 2^{n}$. In other words, $\mathrm{Iso}\, (X_n)$ as a subgroup of $\mathrm{Iso}\, (X_{n+1})$ is acting on the “upper” hyperface $V_n\times \{1\}$ of the cube $V_{n+1}=\{0,1\}^{n+1}=V_n\times \{0,1\}$ in the same way as it acts on the “lower” hyperface $V_n\times \{0\}\cong V_n$. Thus, we have the ascending chain

$$\begin{aligned} \mathrm{Iso}\, (X_{n_0})<\cdots< \mathrm{Iso}\, (X_{n})<\mathrm{Iso}\, (X_{n+1})<\cdots \end{aligned}$$

and the corresponding ascending chain of symmetric subgroups

$$\begin{aligned} S_{n_0}<\cdots< S_{n}<S_{n+1}<\cdots . \end{aligned}$$

For a fixed $w\ge 0$, consider a sequence of landscapes $({\varvec{w}}^{(n)})_{n\ge n_0}$ such that $w_k^{(n)}=w+s$ if $k\in A_n$ and $w_k^{(n)}=w$ otherwise. The sequence $(A_n)_{n=n_0}^\infty$ and the parameters w, s, and $u=w/s$ define the corresponding family of leading eigenvalues $\overline{w}^{(n)}=\overline{w}^{(n)}(q)$, which are solutions of (3), and the family $\overline{u}^{(n)}=\overline{u}^{(n)}(q)$, such that $\overline{u}^{(n)}=\overline{w}^{(n)}/s$.

In Semenov et al. (2014), it was proved that for any $n\ge n_0$ the function $\overline{u}^{(n)}(q)$ has the following properties:

• 1. The function $\overline{u}^{(n)}(q)$ increases on the segment [0.5, 1] and is convex (downward) there.

• 2. $\overline{u}^{(n)}(0.5)=u+\displaystyle {\frac{|A_n|}{2^n}},\quad \overline{u}^{(n)}(1)=u+1$.

Definition 7.3

A sequence $(A_n)_{n=n_0}^\infty$ of $G_n$-orbits is called admissible if the corresponding sequence of values of polynomials $F_{A_n}(2q-1)$ in (23) is non-increasing for each $q\in [0.5,1]$:

$$\begin{aligned} F_{A_n}(2q-1)= & {} \sum _{d=0}^n f^{(n)}_{d}\,(1-q)^dq^{n-d}\ge \sum _{d=0}^{n+1} f^{(n+1)}_{d}\,(1-q)^dq^{n+1-d}\nonumber \\= & {} F_{A_{n+1}}(2q-1),\;n\ge n_0. \end{aligned}$$

(90)

Definition 7.4

A sequence $(A_n)_{n=n_0}^\infty$ of $G_n$-orbits is called a sequence of the moderate growth if

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{|A_n|}{2^n}=0,\quad \text{ or }\quad |A_n|=o(2^n),\quad n\rightarrow \infty . \end{aligned}$$

(91)

The following three lemmas and corollary provide the full proof that all the examples in Sect. 5 deal with admissible sequences of orbits of the moderate growth.

Lemma 7.5

Let $A\subset X_{n_0}$ be a fixed G-orbit. Consider the constant sequences $A_n\equiv A$ and $G_n\equiv G$, $n\ge n_0$. Then the sequence $(A_n)_{n=n_0}^\infty$ is admissible.

Proof

Since the orbit is not changing as $n\rightarrow \infty$, then it follows from (23) that

$$\begin{aligned} F_{A_n}(2q-1)=q^{n-n_0}F_{A_{n_0}}(2q-1),\quad q\in [0,1]. \end{aligned}$$

The polynomial $F_{A_n}(2q-1)>0$ and $q^{n-n_0}\ge q^{n+1-n_0}$ on [0.5, 1]. Hence, (90) holds. $\square$

Lemma 7.6

Let $a_n\in X_n$, $a_n^*=2^n-1-a_n$, and $A_n=\{a_n,a_n^*\}$. Let $G_n=G=\{1,g\}$ be the group of order 2 such that $g(a)=a^*$ for any $a\in X_n$. Then the sequence $(A_n)_{n=n_0}^\infty$ is admissible.

Proof

In view of (27) and (47)

$$\begin{aligned} F_{A_n}(2q-1)=q^n+(1-q)^n\ge q^{n+1}+(1-q)^{n+1}=F_{A_{n+1}}(2q-1),\quad q\in [0,1]. \end{aligned}$$

$\square$

Lemma 7.7

Let p be a fixed number, $n\ge n_0=2p$. Let $A_n=A_{n,p}=\{a\in X_n\,|\,H_a=p\}$. Then $(A_n)_{n=n_0}^\infty$ is an admissible sequence.

Proof

It follows from (45) that

$$\begin{aligned} F_{A_n}(2q-1)=\sum \limits _{k=0}^p {\left( {\begin{array}{c}p\\ k\end{array}}\right) }{\left( {\begin{array}{c}n-p\\ k\end{array}}\right) }(1-q)^{2k}q^{n-2k}=:F_{n,p}(q)\;,\quad 0\le p\le n. \end{aligned}$$

(92)

At the same time, consider the polynomials

$$\begin{aligned} G_{n,p}(q):=\sum \limits _{k=1}^p {\left( {\begin{array}{c}p\\ k\end{array}}\right) }{\left( {\begin{array}{c}n-p\\ k-1\end{array}}\right) }(1-q)^{2k}q^{n+1-2k}. \end{aligned}$$

(93)

By definition, $F_{0,0}(q)\equiv 1$, $G_{n,0}(q)=0$. Applying the binomial formulas ${\left( {\begin{array}{c}n+1-p\\ k\end{array}}\right) }={\left( {\begin{array}{c}n-p\\ k\end{array}}\right) }+{\left( {\begin{array}{c}n-p\\ k-1\end{array}}\right) }$ to (92) and ${\left( {\begin{array}{c}p\\ k\end{array}}\right) }={\left( {\begin{array}{c}p-1\\ k\end{array}}\right) }+{\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right) }$ to (93), we get the following recursive relations:

$$\begin{aligned} F_{n+1,p}(q)=qF_{n,p}(q)+G_{n,p}(q),\quad G_{n+1,p}(q)=(1-q)^2 F_{n,p-1}(q)+qG_{n,p-1}(q). \end{aligned}$$

(94)

When we substitute the left-hand side of the second formula (94) into the first one (with the change $n\rightarrow n-1$) and then iterate such substitutions, we get

$$\begin{aligned} F_{n+1,p}(q)=qF_{n,p}(q)+(1-q)^2\sum _{j=1}^p q^{j-1}F_{n-j,p-j}(q)\;. \end{aligned}$$

(95)

In the same way, the equality

$$\begin{aligned} G_{n+1,p}(q)=(1-q)^2\sum _{j=1}^p q^{j-1}F_{n+1-j,p-j}(q) \end{aligned}$$

(96)

can be obtained.

Formulas (94) imply also that

$$\begin{aligned} F_{n,p}(q)-F_{n+1,p}(q)= & {} (1-q)F_{n,p}(q)-G_{n,p}(q)\\= & {} (1-q)qF_{n-1,p}(q)+(1-q)G_{n-1,p}(q)-G_{n,p}(q), \end{aligned}$$

or

$$\begin{aligned} F_{n,p}(q)-F_{n+1,p}(q)= & {} q\left( (1-q)F_{n-1,p}(q)-G_{n,p}(q)\right) \nonumber \\&+\, (1-q)\left( G_{n-1,p}(q)-G_{n,p}(q)\right) . \end{aligned}$$

(97)

Our objective is to prove that

$$\begin{aligned} F_{2p+k+1,p}(q)\le F_{2p+k,p}(q),\qquad k\ge 0,\quad q\in [0,1]. \end{aligned}$$

We will proceed by induction on p and, for a fixed p, by induction on k.

First of all, the case $p=0$ is trivial since $F_{n,0}(q)=q^n$.

Let $p\ge 1$ be fixed and let $k=0$. Substituting $n=2p$ into (97), we get

$$\begin{aligned} F_{2p,p}(q)-F_{2p+1,p}(q)= & {} q\left( (1-q)F_{2p-1,p}(q)-G_{2p,p}(q)\right) \nonumber \\&+\, (1-q)\left( G_{2p-1,p}(q)-G_{2p,p}(q)\right) . \end{aligned}$$

Let us show that both summands in the right-hand side are nonnegative on [0, 1]. On the one hand, by definition we have $F_{n,p}(q)=F_{n,n-p}(q)$. Then in view of (94)

$$\begin{aligned}&(1-q)F_{2p-1,p}(q)-G_{2p,p}(q)=(1-q)F_{2p-1,p-1}(q)-G_{2p,p}(q)\\&\quad =(1-q)F_{2p-1,p-1}(q)-((1-q)^2F_{2p-1,p-1}(q)+qG_{2p-1,p-1}(q))\\&\quad =q(1-q)F_{2p-1,p-1}(q)-qG_{2p-1,p-1}(q)\\&\quad =q(1-q)F_{2p-1,p-1}(q)-q(F_{2p,p-1}-qF_{2p-1,p-1}(q))\\&\quad =q(F_{2p-1,p-1}(q)-F_{2p,p-1}(q))\ge 0\;,\quad q\in [0,1], \end{aligned}$$

by the inductive hypothesis.

On the other hand, it follows from (96) that

$$\begin{aligned} G_{2p-1,p}(q)-G_{2p,p}(q)=(1-q)^2\sum _{j=1}^p q^{j-1}(F_{2p-j,p-j}(q)-F_{2p+1-j,p-j}(q)) \ge 0 \end{aligned}$$

on [0, 1] by the same reasons. This finishes the proof for the case $k=0$.

Let $k\ge 1$. Then by virtue of (95), we can assert that

$$\begin{aligned}&F_{2p+k,p}(q)-F_{2p+k+1,p}(q)\\&=q(F_{2p+k-1,p}(q)-F_{2p+k,p}(q))+(1-q)^2\sum _{j=1}^p q^{j-1}(F_{2p+k-1-j,p-j}(q)\nonumber \\&-F_{2p+k-j,p-j}(q))\ge 0 \end{aligned}$$

on [0, 1] by the inductive hypothesis. The lemma is proved. $\square$

Corollary 7.8

Let $A_{2n}=A_{2n,n}=\{a\in X_{2n}\,|\,H_a=n\}$. Then $(A_{2n})_{n=n_0}^\infty$ is an admissible sequence.

Proof

In the notation of Lemma 7.7, let us prove that $F_{2n,n}(q)\le F_{2n-2,n-1}(q)$ on [0, 1]. From the first formula (94), we can find the expressions $G_{n+1,p}(q)=F_{n+2,p}(q)-qF_{n+1,p}(q)$, $G_{n,p-1}(q)=F_{n+1,p-1}(q)-qF_{n,p-1}(q)$ and substitute them into the second one. The simplification yields

$$\begin{aligned} F_{n+2,p}(q)=(1-2q)F_{n,p-1}(q)+qF_{n+1,p}(q)+qF_{n+1,p-1}(q). \end{aligned}$$

Consequently, choosing appropriate values for n, p in this formula, we get

$$\begin{aligned} F_{2n-2,n-1}(q)-F_{2n,n}(q)= & {} q(F_{2n-2,n-1}(q)-F_{2n-1,n}(q))\\&+\, q(F_{2n-2,n-1}(q)-F_{2n-1,n-1}(q)). \end{aligned}$$

But in view of (92) $F_{n,p}(q)=F_{n,n-p}(q)$ for all n and p, $0\le p\le n$. Hence $F_{2n-1,n}(q)=F_{2n-1,n-1}(q)$ and it follows from Lemma 7.7 that

$$\begin{aligned} F_{2n-2,n-1}(q)-F_{2n,n}(q)=2q(F_{2n-2,n-1}(q)-F_{2n-1,n-1}(q))\ge 0\, \end{aligned}$$

on the segment [0, 1]. $\square$

Consider a sequence $(A_n)_{n=n_0}^\infty$ of $G_n$-orbits. Our next aim is to investigate what happens with the corresponding family $(\overline{u}^{(n)})_{n=n_0}^\infty$ as $n\rightarrow \infty$.

Proposition 7.9

If $(A_n)_{n=n_0}^\infty$ is an admissible sequence of $G_n$-orbits, then for each fixed $q\in [0.5,1]$ the sequence $(\overline{u}^{(n)}(q))_{n=n_0}^\infty$ is a non-increasing sequence as $n\rightarrow \infty$. If, additionally, $(A_n)_{n=n_0}^\infty$ is a sequence of the moderate growth, then $\lim \limits _{n\rightarrow \infty }\overline{u}^{(n)}(0.5)=u$ and $\lim \limits _{n\rightarrow \infty }\overline{u}^{(n)}(1)=u+1$.

Proof

The second assertion follows directly from Property 2 of $\overline{u}^{(n)}(q)$ above. Let us proof the first one. Equation (82) for $u\ne 0$ can be rewritten in the form

$$\begin{aligned} u= & {} \frac{u}{\overline{u}^{(n)}(q)}\overline{u}^{(n)}(q)= \sum _{c=0}^{\infty }\left( \frac{u}{\overline{u}^{(n)}(q)}\right) ^{c+1} F_{A_n}((2q-1)^{c+1})\nonumber \\= & {} \sum _{m=1}^{\infty }\left( \frac{u}{\overline{u}^{(n)}(q)}\right) ^{m} F_{A_n}((2q-1)^m). \end{aligned}$$

(98)

It follows from Definition 7.3 that at each fixed point $q\in [0.5,1]$ the sequence of positive coefficients $(F_{A_n}((2q-1)^{c+1}))_{n\ge n_0}$ is non-increasing for any $c+1\in \mathbf {N}$. But the left-hand side u of (98) is constant. This implies that $(\overline{u}^{(n)}(q))_{n\ge n_0}$ must be a non-increasing sequence for each $q\in [0.5,1]$. $\square$

Hence we can conclude that the curve $\overline{u}=\overline{u}^{(n+1)}(q)$ always passes under the curve $\overline{u}=\overline{u}^{(n)}(q)$ in the rectangle $\{0.5\le q\le 1,\;u\le \overline{u}\le u+1\}$ if $(A_n)_{n=n_0}^\infty$ is an admissible sequence of $G_n$-orbits, see Fig. 5.

Proposition 7.9 and Property 1 of $\overline{u}^{(n)}(q)$ yield

Corollary 7.10

If $(A_n)_{n=n_0}^\infty$ is an admissible sequence of $G_n$-orbits of the moderate growth, then for any fixed $\varepsilon \in (0,1]$ there exists $N_0\in \mathbf {N}$ such that for any $n\ge N_0$ the curve $\overline{u}=\overline{u}^{(n)}(q)$ intersects the line $\overline{u}=u+\varepsilon$ at a unique point $q^{(n)}(\varepsilon ,u)\in (0.5,1]$.

Note that by virtue of (82), (98), and (26) the value $q^{(n)}(\varepsilon ,u)$ from the previous corollary can be found from one of the following equations

$$\begin{aligned} u+\varepsilon = \sum _{c=0}^{\infty }\left( \frac{u}{u+\varepsilon }\right) ^{c} F_{A_n}((2q-1)^{c+1}), \end{aligned}$$

(99)

or,

$$\begin{aligned} \sum _{d=0}^N \frac{h_d(2q-1)^d}{u+\varepsilon -u(2q-1)^d}=1. \end{aligned}$$

(100)

Another almost immediate result is given in the following

Proposition 7.11

If $(A_n)_{n\ge N_0}$ is an admissible sequence of $G_n$-orbits of the moderate growth, then for fixed $(\varepsilon ,u)$ the sequence $(q^{(n)}(\varepsilon ,u))_{n\ge N_0}$ is non-decreasing as $n\rightarrow \infty$ and the inequality

$$\begin{aligned} q^{(n)}(\varepsilon ,0)\le q^{(n)}(\varepsilon ,u)\le \root n \of {\frac{u+\varepsilon }{u+1}}\le 1-\frac{1-\varepsilon }{n(u+1)}\, \end{aligned}$$

(101)

holds.

Proof

The upper bound (see Sect. 7.1) $\overline{u}=u^{(n)}_{up,1}(q)= u+ F_{A_n}(2q-1)$ gives rise to the lower bound in (101) since the equation $u+\varepsilon =u+ F_{A_n}(2q-1)$ is equivalent to (99) when $u=0$. The lower bound (87) $\overline{u}=(u+1)q^n$ provides the upper bound in (101).

Since $\overline{u}=(u+1)q^n$ is convex downward, if $q\in [0.5,1]$ and $\overline{u}=u+1-n(u+1)(1-q)$ is the equation of the tangent at $q=1$ to the curve $\overline{u}=(u+1)q^n$ then we get the last inequality in (101). Note that the curve $\overline{u}=\overline{u}^{(n)}(q)$ has the same tangent at $q=1$ [see, for instance, Semenov et al. (2014)]. $\square$

The obtained results are illustrated in Fig. 5.

Fig. 5

(Color figure online) The curves in the coordinates $q,\overline{u}$ defined by (from top to bottom): $\overline{u}=u+F_{A_n}(2q-1),\,\overline{u}=\overline{u}^n(q),\,\overline{u}=\overline{u}^{n+1}(q),\,\overline{u}=(u+1)q^{n+1}$. The points of intersections of these curves with the dotted line $\overline{u}=u+\varepsilon$ define the values $q^{(n)}(\varepsilon ,0),\,q^{(n)}(\varepsilon ,u),\,q^{(n+1)}(\varepsilon ,u),\,1-\frac{1-\varepsilon }{(n+1)(u+1)}$ respectively, see also (101)

1.3 Threshold-Like Behavior

In this subsection, we define what we call the threshold-like behavior and provide sufficient conditions for the sequences of admissible orbits to possess this kind of behavior. The main conclusion, which can be stated in a form of a conjecture, emphasizes the role of geometry for the threshold-like behavior to occur. Loosely speaking, if the admissible sequence of orbits “looks like a point” asymptotically, i.e., basically indistinguishable from the single-peaked landscape in the infinite length limit, then the threshold-like behavior is observed. This conjecture is supported by the analysis in Corollary 7.16. We conjecture, as numerical experiments show, that the opposite is true: If asymptotically the admissible sequence of orbits is different from a point, then there exists no threshold-like behavior. We also present formula (108), which gives the estimate of the critical mutation rate for our one orbit quasispecies models. In the more general situations, such as for mesa landscapes [see discussion in Sect. 2 or Wolff and Krug (2009)], we expect that the formula will be different, but the general qualitative conjecture still holds.

Let us introduce the notation

$$\begin{aligned} q^{(n)}_*(\varepsilon ,u)=\root n \of {\frac{u+\varepsilon }{u+1}}, \end{aligned}$$

(102)

from where

$$\begin{aligned} \lim _{n\rightarrow \infty }n(1-q^{(n)}_*(\varepsilon ,u))=\log \frac{u+1}{u+\varepsilon }. \end{aligned}$$

(103)

It follows that for a fixed $u>0$

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\lim _{n\rightarrow \infty }n(1-q^{(n)}_*(\varepsilon ,u))=\log \frac{u+1}{u}. \end{aligned}$$

(104)

It is known [e.g., Semenov et al. (2014)] that for the single-peaked landscape the curve $\overline{u}=\overline{u}^{(n)}(q)$ passes very close to the lower bound $\overline{u}=\max \{u,(u+1)q^n\}$ in such a way that

$$\begin{aligned} q^{(n)}(\varepsilon ,u)=q^{(n)}_*(\varepsilon ,u)-o\left( \frac{1}{n}\right) = \root n \of {\frac{u+\varepsilon }{u+1}}-o\left( \frac{1}{n}\right) \end{aligned}$$

as $n\rightarrow \infty$ (from (101) we have the inequality $q^{(n)}(\varepsilon ,u)\le q^{(n)}_*(\varepsilon ,u)$).

Our next aim is to investigate what happens with the curve $\overline{u}=\overline{u}^{(n)}(q)$ as $n\rightarrow \infty$. It is more conveniently done in coordinates x, L, defined by

$$\begin{aligned} q=1-\frac{x}{n},\quad 0\le x\le \frac{n}{2},\quad \overline{u}=(u+1)e^{-L},\quad 0\le L\le \log \frac{u+1}{u}. \end{aligned}$$

(105)

We will assume that $u>0$ in (105). Hence the curve $\overline{u}=\overline{u}^{(n)}(q)$ transforms into the curve

$$\begin{aligned} L_n(x)=\log (u+1)-\log \overline{u}^{(n)}\left( 1-\frac{x}{n}\right) . \end{aligned}$$

(106)

Note that $L_n(0)=0$ since ${u}^{(n)}(1)=u+1$ for any n.

Definition 7.12

We say that an admissible sequence $(A_n)_{n\ge n_0}$ of the moderate growth, or, equivalently, the family $(\overline{u}^{(n)})_{n\ge n_0}$ possesses the threshold-like behavior on the segment [0.5, 1] if for each fixed $x\ge 0$ and the corresponding functions $L_n(x)$ it is true that

$$\begin{aligned} \lim _{n\rightarrow \infty } L_n(x)=L(x)={\left\{ \begin{array}{ll}x,&{}0\le x<\log \frac{u+1}{u},\\ \log \frac{u+1}{u},&{}x\ge \log \frac{u+1}{u}. \end{array}\right. } \end{aligned}$$

(107)

The definition above is illustrated in Fig. 6.

Fig. 6

(Color figure online) The limit function L(x) in Definition 7.12 of the threshold-like behavior

If the threshold-like behavior is present in the two-valued fitness landscape, then the following formula provides an approximation for the threshold mutation rate $q_*^{(n)}(u)$, $n\gg 1$:

$$\begin{aligned} q^{(n)}_*(u)\approx 1-\frac{1}{n}\log \frac{u+1}{u}=1-\frac{1}{n}\log \frac{w+s}{w}, \end{aligned}$$

(108)

which, of course, coincides with the classical estimate for the error threshold for the single-peaked landscape (Eigen et al. 1988; Semenov et al. 2014).

If the sequence of continuous functions $(\overline{u}^{(n)})_{n\ge n_0}$ has the threshold-like behavior, then it converges not uniformly on [0.5, 1], as $n\rightarrow \infty ,$ to the discontinuous function $\psi (q)$ such that $\psi (q)=u$ if $0.5\le q<1$ and $\psi (1)=u+1$.

The following theorems and corollaries provide sufficient conditions under which an admissible sequence of orbits of the moderate growth shows the threshold-like behavior.

Theorem 7.13

In the above notation, suppose that for $n\ge n_0$ an admissible sequence of $G_n$-orbits $A_n\subset X_n$ (${G}_n\leqslant \mathrm{Iso}\, (X_n)$) of the moderate growth is given and $u>0$. Suppose also that for $n\ge n_0$ the inequality

$$\begin{aligned} F_{A_n}(2q-1)\le (2q-1)^{n/2}+M_n,\quad 0.5\le q\le 1 , \end{aligned}$$

(109)

is satisfied for some constants $M_n$ such that $\lim \limits _{n\rightarrow \infty }M_n=0$. Then the sequence $(A_n)_{n\ge n_0}$ shows the threshold-like behavior on the segment [0.5, 1].

Proof

In view of Eq. (82), in coordinates x, L:

$$\begin{aligned} (u+1)e^{-L_n(x)}= \sum _{c=0}^{\infty }\left( \frac{u}{(u+1)e^{-L_n(x)}}\right) ^c F_{A_n}\left( \left( 1-\frac{2x}{n}\right) ^{c+1}\right) , \end{aligned}$$

therefore (putting $m=c+1$)

$$\begin{aligned} u= \sum _{m=1}^{\infty }\left( \frac{u}{u+1}\right) ^m e^{mL_n(x)}F_{A_n}\left( \left( 1-\frac{2x}{n}\right) ^{m}\right) . \end{aligned}$$

(110)

The lower bound (87), $\overline{u}\ge (u+1)q^n$, implies for $x\in \left[ 0,\log \frac{u+1}{u}\right)$, $n\gg 1$,

$$\begin{aligned} L_n(x)=\log \frac{u+1}{\overline{u}^{(n)}(1-\frac{x}{n})}\le -n\log \left( 1-\frac{x}{n}\right) . \end{aligned}$$

(111)

Consequently, we have on $\left[ 0,\log \frac{u+1}{u}\right)$

$$\begin{aligned} \limsup _{n\rightarrow \infty } L_n(x)\le -\lim _{n\rightarrow \infty }n\log \left( 1-\frac{x}{n}\right) = x. \end{aligned}$$

(112)

On the other hand, the function $F_{A_n}(2q-1)$, as the leading eigenvalue for $u=0$ [see (27)], is increasing on the segment [0.5, 1]. In view of the inequality $1-t\le e^{-t}$, we have

$$\begin{aligned} F_{A_n}\left( \left( 1-\frac{2x}{n}\right) ^{m}\right) \le F_{A_n}\left( e^{-2mx/n}\right) . \end{aligned}$$

Make the substitution $2q-1=e^{-2x/n}$ into (109), where $0\le x<+\infty$. Then the following inequality

$$\begin{aligned} e^{mL_n(x)}F_{A_n}\left( \left( 1-\frac{2x}{n}\right) ^{m}\right) \le e^{mL_n(x)}F_{A_n}\left( e^{-2mx/n}\right) \le e^{m(L_n(x)-x)}+M_ne^{mL_n(x)}\, \end{aligned}$$

holds. Hence, (110) yields

$$\begin{aligned} u\le \sum _{m=1}^{\infty }\left( \frac{u}{u+1}\right) ^m e^{m(L_n(x)-x)}+M_n\sum _{m=1}^{\infty }\left( \frac{ue^{L_n(x)}}{u+1}\right) ^m . \end{aligned}$$

In view of (111), both progressions in the right-hand side converge for $x\in \left[ 0,\log \frac{u+1}{u}\right)$ and $n\gg 1$.

The simplification provides the inequality

$$\begin{aligned} 1\le e^{L_n(x)-x}+ M_n\frac{e^{L_n(x)}(u+1-ue^{L_n(x)-x})}{(u+1)(u+1-ue^{L_n(x)})}. \end{aligned}$$

Since $M_n\rightarrow 0$ and the inequality (111) holds, we get finally $\liminf \limits _{n\rightarrow \infty } e^{L_n(x)-x}\ge 1\,$, or,

$$\begin{aligned} \liminf _{n\rightarrow \infty } L_n(x)\ge x. \end{aligned}$$

(113)

It follows from (112), (113) that $\lim \limits _{n\rightarrow \infty }L_n(x)=x$ if $0\le x< \log \frac{u+1}{u}$. The increasing functions $L_n(x)$ cannot exceed the value $\log \frac{u+1}{u}$. Thus, the threshold-like behavior is observed. $\square$

Corollary 7.14

The sequence of constant single-peaked landscapes $A_n\equiv \{a\},\,n\ge n_0$, possesses the threshold-like behavior.

Proof

Condition (109) of Theorem 7.13 reads as follows: The inequality

$$\begin{aligned} q^n\le (2q-1)^{n/2}+M_n,\quad 0.5\le q\le 1, \end{aligned}$$

(114)

holds for some constants $M_n$ such that $\lim \limits _{n\rightarrow \infty }M_n=0$.

Let us show that the constants

$$\begin{aligned} M_n=\frac{1}{n}\left( 1-\frac{1}{n}\right) ^{n-1}\le \frac{1}{e(n-1)} \end{aligned}$$

fit. Consider the auxiliary function $\varphi _n(q)=q^n- (2q-1)^{n/2}$. Its maximum $\mu _n$ on the segment [0.5, 1] is reached at some point $q_n<1$. This point is a root of the equation

$$\begin{aligned} \varphi '_n(q)=nq^{n-1}-n (2q-1)^{n/2-1}=0\;, \quad \text{ or }\;\;(2q-1)^{n/2}=(2q-1)q^{n-1}. \end{aligned}$$

Hence, by the definition of $\varphi _n(q)$, we get

$$\begin{aligned} \mu _n=\varphi _n(q_n)=q_n^n-(2q_n-1)q_n^{n-1}=q_n^{n-1}(1-q_n). \end{aligned}$$

The function $M(t)=t^{n-1}(1-t)$ achieves its maximum on [0,1], which is equal to $M_n=\frac{1}{n}\left( 1-\frac{1}{n}\right) ^{n-1}$, at $t_n=1-\frac{1}{n}$. Hence, $\mu _n\le M_n$ and the conditions of Theorem 7.13 hold. $\square$

Theorem 7.13 and Corollary 7.14 are the key results as the following theorem shows. We are convinced that the reason for the error threshold effect is geometric. More precisely, in view of (23) the polynomial $F_{A_n}(2q-1)$ can be always represented in the form

$$\begin{aligned} F_{A_n}(2q-1)=q^n+\sum _{k=1}^n f_k^{(n)}\,(1-q)^{k}q^{n-k}, \end{aligned}$$

(115)

where $f_k^{(n)}=\#\{b\in A_n\,|\,H_{ab}=k\}$. Thus, this polynomial can be viewed as a kind of the spherical growth function of the orbit $A_n$ with respect to an arbitrary fixed point $a\in A_n$.

Theorem 7.15

In the above notation, assume that for any $n\ge n_0$ an admissible sequence of $G_n$-orbits $A_n\subset X_n$ ($G_n\leqslant \mathrm{Iso}\,( X_n)$), $n\ge n_0$ of the moderate growth is given and $u>0$. If either

$$\begin{aligned} \lim _{n\rightarrow \infty }\max _{q\in [0.5,1]}\sum _{k=1}^{\lfloor n/2\rfloor } f_k^{(n)}\,(1-q)^{k}q^{n-k}=0, \end{aligned}$$

(116)

or

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{k=1}^{\lfloor n/2\rfloor } f_k^{(n)}\,\left( \frac{k}{n}\right) ^{k}\left( 1-\frac{k}{n}\right) ^{n-k}=0, \end{aligned}$$

(117)

then the sequence $(A_n)_{n\ge n_0}$ possesses the threshold-like behavior.

Proof

The polynomial $P_k^{(n)}(q)=(1-q)^kq^{n-k}$ decreases on the segment [0.5,1] if $n<2k\le 2n$ and achieves its maximal value $2^{-n}$ at $q=0.5$. If $1\le k\le \lfloor n/2\rfloor$ then the maximal value of $P_k^{(n)}(q)=(1-q)^kq^{n-k}$ on [0.5,1] is achieved at the point $q_k^{(n)}=1-\frac{k}{n}$ and is equal to $\left( \frac{k}{n}\right) ^{k}\left( 1-\frac{k}{n}\right) ^{n-k}$.

Denote $F_n=\max \limits _{q\in [0.5,1]}\sum _{k=1}^{\lfloor n/2\rfloor }f_k^{(n)}(1-q)^{k}q^{n-k}$. Then Corollary 7.14, (114), and (115) together yield

$$\begin{aligned} \begin{aligned} F_{A_n}(2q-1)&=q^n+\sum _{k=1}^n f_k^{(n)}(1-q)^{k}q^{n-k} \le q^n+F_n+ \sum _{k=\lfloor n/2\rfloor +1}^n \frac{f_k^{(n)}}{2^n}\\&\le (2q-1)^{n/2}+\frac{1}{n}\left( 1-\frac{1}{n}\right) ^{n-1}\\&+F_n+\frac{|A_n|}{2^n}=(2q-1)^{n/2}+o(1),\quad n\rightarrow \infty . \end{aligned} \end{aligned}$$

(118)

On the other hand, if the equality (117) holds, we can substitute $\sum _{k=1}^{\lfloor n/2\rfloor } f_k^{(n)}\,\left( \frac{k}{n}\right) ^{k}\left( 1-\frac{k}{n}\right) ^{n-k}$ for $F_n$ since $F_n\le \sum _{k=1}^{\lfloor n/2\rfloor } f_k^{(n)}\left( \frac{k}{n}\right) ^{k}\left( 1-\frac{k}{n}\right) ^{n-k}$. Hence, Theorem 7.13 implies that the sequence $(A_n)_{n\ge n_0}$ shows the threshold-like behavior. $\square$

Corollary 7.16

The following sequences of orbits possess the threshold-like behavior:

• (i) All the constant sequences $A_n\equiv A$ (see, for instance, Example 5.6 of the quaternion landscape);

• (ii) All the antipodal sequences $A_n= \{a,a^*\}\subset X_n$ (see Example 5.7);

• (iii) All the permutation invariant sequences $A_{p,n}$ where $A_{p,n}=\{a\in X\,|\,H_a=p\}\;$, $p=0,1,\dots ,n\,$, and p does not depend on $n\ge p$ (see Examples 5.1 and 5.5).

Proof

(i) Let $A\subset X_{n_0}$. Since the orbit is fixed then for $n\ge n_0$ all the coefficients $f_k^{(n)}\equiv f_k^{(n_0)}$ and $f_k^{(n)}\equiv 0$ when $k>n_0$. It follows that the assumption (117) of Theorem 7.15 that

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{k=1}^{\lfloor n_0/2\rfloor } f_k^{(n_0)}\,\left( \frac{k}{n}\right) ^{k}\left( 1-\frac{k}{n}\right) ^{n-k}=0\, \end{aligned}$$

holds since $k\le \lfloor n_0/2\rfloor$ and, consequently, $\left( \frac{k}{n}\right) ^{k}\rightarrow 0$ as $n\rightarrow \infty$ provided $\left( 1-\frac{k}{n}\right) ^{n-k}\le 1$.

(ii) In this case, $F_{A_n}(q)=q^n+(1-q)^n$. Then $f_k^{(n)}=0$, $k=1,\dots , \lfloor n/2\rfloor$. Then both assumptions (116), (117) hold.

(iii) We may suppose that $n\ge 2p=n_0$. In view of (29)

$$\begin{aligned} F_{A_{p,n}}(q)=\sum \limits _{k=0}^p {\left( {\begin{array}{c}p\\ k\end{array}}\right) }{\left( {\begin{array}{c}n-p\\ k\end{array}}\right) }(1-q)^{2k}q^{n-2k}. \end{aligned}$$

Hence, since p is fixed, ${\left( {\begin{array}{c}p\\ k\end{array}}\right) }<2^p$, $1-\frac{2k}{n}\le 1$:

$$\begin{aligned}&\sum \limits _{k=1}^p {\left( {\begin{array}{c}p\\ k\end{array}}\right) }{\left( {\begin{array}{c}n-p\\ k\end{array}}\right) }\left( \frac{k}{n}\right) ^{2k}\left( 1-\frac{2k}{n}\right) ^{n-2k}\\&\quad \le (2p)^{2p}\sum \limits _{k=1}^p \frac{1}{n^{2k}}\,{\left( {\begin{array}{c}n-p\\ k\end{array}}\right) }\le (2p)^{2p}\sum \limits _{k=1}^p \frac{n^k}{k!\,n^{2k}}=o(1) \end{aligned}$$

as $n\rightarrow \infty$. Consequently, the condition (117) is satisfied. $\square$

Remark 7.17

In contrast, if $A_{2n}=A_{n,2n}$ is the sequence of the fitness landscapes in Example 5.6, then the numerical calculations (Fig. 7) show that this sequence does not demonstrate the threshold-like behavior. The approximate formula (46) provides a lower bound

$$\begin{aligned}&\max _{q\in [0.5,1]}\sum _{k=1}^{n} {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2\,(1-q)^{2k}q^{2n-2k}\ge 0.183\approx \frac{4r-1}{8r\sqrt{\pi r}},\\&\quad r=-\frac{3}{4}W\left( -\frac{1}{3\root 3 \of {\pi }}\right) \approx 1.7423, \end{aligned}$$

where W(z) is (a branch of) the Lambert W function ($W(z)e^{W(z)}=z$). Hence, the sufficient conditions for the threshold-like behavior are not satisfied in this case.

Fig. 7

(Color figure online) The leading eigenvalue $\overline{w}(q)$ versus the mutation rate q in the case of the fitness landscape in Example 5.6. The sequence length $2n=50$ in the left panel and $2n=100$ in the right panel. Note the absence of the threshold-like behavior

A natural question to ask is whether the given sufficient conditions are also necessary for the threshold-like behavior. While at this point we do not have a full answer for this question, we can present a sufficient condition for the absence of the threshold-like behavior of the sequence $(\overline{u}^{(n)})_{n\ge n_0}$ as $n\rightarrow \infty$. This sufficient condition shows in a way that the condition (116) is “almost” necessary for the error threshold.

Proposition 7.18

Suppose that there exist constants $\varepsilon >0$, $x>0$ such that for all $n\gg n_0$ the inequality

$$\begin{aligned} F_{A_n}(2q_n-1)\ge (u+1)(q_n^n+2\varepsilon ),\quad q_n=1-\frac{x}{n}, \end{aligned}$$

(119)

holds for sufficiently small $u>0$. Then the sequence $(A_n)_{n\ge n_0}$ possesses no threshold-like behavior.

Proof

We can assume that $x<\log \frac{u+1}{u}$ for sufficiently small $u>0$ and $q_n>0.5$ for sufficiently large n. In view of (82)

$$\begin{aligned} \overline{u}^{(n)}(q_n)= & {} F_{A_n}(2q_n-1)+\sum _{c=1}^{\infty }\left( \frac{u+1}{\overline{u}^{(n)}(q_n)}\right) ^c F_{A_n}\left( (2q_n-1)^{c+1}\right) \nonumber \\\ge & {} F_{A_n}(2q_n-1)\ge (u+1)(q_n^n+2\varepsilon ). \end{aligned}$$

Hence,

$$\begin{aligned} L_n(x)= & {} \log (u+1)-\log \overline{u}^{(n)}(q_n)\le -\log (q_n^n+2\varepsilon )\nonumber \\= & {} -\log \left( \left( 1-\frac{x}{n}\right) ^n+2\varepsilon \right) <-\log (e^{-x}+\varepsilon ). \end{aligned}$$

for $n\gg n_0$. Consequently, $\limsup \limits _{n\rightarrow \infty } L_n(x)\le -\log (e^{-x}+\varepsilon )<x$. $\square$

Note that in Remark 7.17 we can take $x=r\approx 1.7423$, $u\le 0.1$, $\varepsilon =0.01$, $n\ge 4$.