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3619 KiB

Open AccessArticle

Stationary Stability for Evolutionary Dynamics in Finite Populations

by Marc Harper and Dashiell Fryer

Entropy 2016, 18(9), 316; https://doi.org/10.3390/e18090316 - 25 Aug 2016

Cited by 10 | Viewed by 7007

Abstract

We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the [...] Read more.

We demonstrate a vast expansion of the theory of evolutionary stability to finite populations with mutation, connecting the theory of the stationary distribution of the Moran process with the Lyapunov theory of evolutionary stability. We define the notion of stationary stability for the Moran process with mutation and generalizations, as well as a generalized notion of evolutionary stability that includes mutation called an incentive stable state (ISS) candidate. For sufficiently large populations, extrema of the stationary distribution are ISS candidates and we give a family of Lyapunov quantities that are locally minimized at the stationary extrema and at ISS candidates. In various examples, including for the Moran and Wright–Fisher processes, we show that the local maxima of the stationary distribution capture the traditionally-defined evolutionarily stable states. The classical stability theory of the replicator dynamic is recovered in the large population limit. Finally we include descriptions of possible extensions to populations of variable size and populations evolving on graphs. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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294 KiB

Open AccessArticle

A Cost/Speed/Reliability Tradeoff to Erasing

by Manoj Gopalkrishnan

Entropy 2016, 18(5), 165; https://doi.org/10.3390/e18050165 - 28 Apr 2016

Cited by 6 | Viewed by 4794

Abstract

We present a Kullback–Leibler (KL) control treatment of the fundamental problem of erasing a bit. We introduce notions of reliability of information storage via a reliability timescale

τ_{r}

, and speed of erasing via an erasing timescale

τ_{e}

. Our problem [...] Read more.

We present a Kullback–Leibler (KL) control treatment of the fundamental problem of erasing a bit. We introduce notions of reliability of information storage via a reliability timescale

τ_{r}

, and speed of erasing via an erasing timescale

τ_{e}

. Our problem formulation captures the tradeoff between speed, reliability, and the KL cost required to erase a bit. We show that rapid erasing of a reliable bit costs at least

log 2 - log (1 - e^{- \frac{τ_{e}}{τ_{r}}}) > log 2

, which goes to

\frac{1}{2} log \frac{2 τ_{r}}{τ_{e}}

when

τ_{r} > > τ_{e}

. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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289 KiB

Open AccessArticle

Open Markov Processes: A Compositional Perspective on Non-Equilibrium Steady States in Biology

by Blake S. Pollard

Entropy 2016, 18(4), 140; https://doi.org/10.3390/e18040140 - 15 Apr 2016

Cited by 9 | Viewed by 5735

Abstract

In recent work, Baez, Fong and the author introduced a framework for describing Markov processes equipped with a detailed balanced equilibrium as open systems of a certain type. These “open Markov processes” serve as the building blocks for more complicated processes. In this [...] Read more.

In recent work, Baez, Fong and the author introduced a framework for describing Markov processes equipped with a detailed balanced equilibrium as open systems of a certain type. These “open Markov processes” serve as the building blocks for more complicated processes. In this paper, we describe the potential application of this framework in the modeling of biological systems as open systems maintained away from equilibrium. We show that non-equilibrium steady states emerge in open systems of this type, even when the rates of the underlying process are such that a detailed balanced equilibrium is permitted. It is shown that these non-equilibrium steady states minimize a quadratic form which we call “dissipation”. In some circumstances, the dissipation is approximately equal to the rate of change of relative entropy plus a correction term. On the other hand, Prigogine’s principle of minimum entropy production generally fails for non-equilibrium steady states. We use a simple model of membrane transport to illustrate these concepts. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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Figure 1

Figure 1
A simple model for passive diffusion across a membrane. Full article ">Figure 2
A depiction of an open Markov process as a labeled, directed graph. Full article ">Figure 3
An open detailed balanced Markov process modeling membrane transport. Full article ">Figure 4
Another layer of membrane whose interior population is labeled by D and whose exterior populations are labeled by <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>′</mo> </msup> </semantics> </math> and E. Full article ">Figure 5
Membranes arranged in series modeled as an open detailed balanced Markov process. Full article ">Figure 6
Composition of open detailed balanced Markov processes results in an open detailed balanced Markov process. Full article ">Figure 7
A depiction of two membranes arranged in series. Full article ">Figure 8
A model of passive transport across a membrane where all transition rates are set equal. Full article ">

331 KiB

Open AccessArticle

The Free Energy Requirements of Biological Organisms; Implications for Evolution

by David H. Wolpert

Entropy 2016, 18(4), 138; https://doi.org/10.3390/e18040138 - 13 Apr 2016

Cited by 20 | Viewed by 10623 | Correction

Abstract

Recent advances in nonequilibrium statistical physics have provided unprecedented insight into the thermodynamics of dynamic processes. The author recently used these advances to extend Landauer’s semi-formal reasoning concerning the thermodynamics of bit erasure, to derive the minimal free energy required to implement an [...] Read more.

Recent advances in nonequilibrium statistical physics have provided unprecedented insight into the thermodynamics of dynamic processes. The author recently used these advances to extend Landauer’s semi-formal reasoning concerning the thermodynamics of bit erasure, to derive the minimal free energy required to implement an arbitrary computation. Here, I extend this analysis, deriving the minimal free energy required by an organism to run a given (stochastic) map π from its sensor inputs to its actuator outputs. I use this result to calculate the input-output map π of an organism that optimally trades off the free energy needed to run π with the phenotypic fitness that results from implementing π. I end with a general discussion of the limits imposed on the rate of the terrestrial biosphere’s information processing by the flux of sunlight on the Earth. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

357 KiB

Open AccessArticle

Maximizing Diversity in Biology and Beyond

by Tom Leinster and Mark W. Meckes

Entropy 2016, 18(3), 88; https://doi.org/10.3390/e18030088 - 9 Mar 2016

Cited by 28 | Viewed by 10473

Abstract

Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or [...] Read more.

Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or lesser importance to rare species. Leinster and Cobbold (2012) proposed a one-parameter family of diversity measures taking into account both this variation and the varying similarities between species. Because of this latter feature, diversity is not maximized by the uniform distribution on species. So it is natural to ask: which distributions maximize diversity, and what is its maximum value? In principle, both answers depend on q, but our main theorem is that neither does. Thus, there is a single distribution that maximizes diversity from all viewpoints simultaneously, and any list of species has an unambiguous maximum diversity value. Furthermore, the maximizing distribution(s) can be computed in finite time, and any distribution maximizing diversity from some particular viewpoint q > 0 actually maximizes diversity for all q. Although we phrase our results in ecological terms, they apply very widely, with applications in graph theory and metric geometry. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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376 KiB

Open AccessArticle

Using Expectation Maximization and Resource Overlap Techniques to Classify Species According to Their Niche Similarities in Mutualistic Networks

by Hugo Fort and Muhittin Mungan

Entropy 2015, 17(11), 7680-7697; https://doi.org/10.3390/e17117680 - 12 Nov 2015

Cited by 3 | Viewed by 5380

Abstract

Mutualistic networks in nature are widespread and play a key role in generating the diversity of life on Earth. They constitute an interdisciplinary field where physicists, biologists and computer scientists work together. Plant-pollinator mutualisms in particular form complex networks of interdependence between often [...] Read more.

Mutualistic networks in nature are widespread and play a key role in generating the diversity of life on Earth. They constitute an interdisciplinary field where physicists, biologists and computer scientists work together. Plant-pollinator mutualisms in particular form complex networks of interdependence between often hundreds of species. Understanding the architecture of these networks is of paramount importance for assessing the robustness of the corresponding communities to global change and management strategies. Advances in this problem are currently limited mainly due to the lack of methodological tools to deal with the intrinsic complexity of mutualisms, as well as the scarcity and incompleteness of available empirical data. One way to uncover the structure underlying complex networks is to employ information theoretical statistical inference methods, such as the expectation maximization (EM) algorithm. In particular, such an approach can be used to cluster the nodes of a network based on the similarity of their node neighborhoods. Here, we show how to connect network theory with the classical ecological niche theory for mutualistic plant-pollinator webs by using the EM algorithm. We apply EM to classify the nodes of an extensive collection of mutualistic plant-pollinator networks according to their connection similarity. We find that EM recovers largely the same clustering of the species as an alternative recently proposed method based on resource overlap, where one considers each party as a consuming resource for the other party (plants providing food to animals, while animals assist the reproduction of plants). Furthermore, using the EM algorithm, we can obtain a sequence of successfully-refined classifications that enables us to identify the fine-structure of the ecological network and understand better the niche distribution both for plants and animals. This is an example of how information theoretical methods help to systematize and unify work in ecology. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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Graphical abstract
Full article ">Figure 1
Four-fold expectation maximization (EM) classification of the unipartite pollinator-pollinator network SCHE. The color of each node i codes the class membership, namely the class r for which <math display="inline"> <msub> <mi>γ</mi> <mrow> <mi>i</mi> <mi>r</mi> </mrow> </msub> </math> is maximum. Since the network contains pollinators with similar pollination preferences, the resulting network structure contains cliques. The EM algorithm uncovers this clique structure, as is evident from the fact that the node colors coincide with the cliques. Full article ">Figure 2
(Top to bottom) 2-, 3- and 4-fold expectation maximization (EM) classification of the SCHE [<a href="#B35-entropy-17-07680" class="html-bibr">35</a>] unipartite pollinator-pollinator network. Each figure is a plot of degree vs. pollinator ID (shown on the x-axis of the bottom figure). The pollinators have been placed according to their ordering on the niche axis, as inferred by the resource overlap (RO) method of <a href="#sec2dot2-entropy-17-07680" class="html-sec">Subsection 2.2</a>. The color of the bars denotes the class that the pollinator has been assigned to by the EM algorithm. The degree refers to the number of different plant species that the pollinator pollinates. A large (low) value of the degree implies that the pollinator is a generalist (specialist). Full article ">Figure 3
(Left) The OFST pollinator-pollinator network with the nine-fold EM classification. The color of each node denotes the class in which the pollinator has been classified by the EM algorithm. (Right, top) The corresponding plot of degree D vs. inferred relative location on a one-dimensional niche axis (legend as in <a href="#entropy-17-07680-f002" class="html-fig">Figure 2</a>). The numbering of the pollinators corresponds to the IDs given in the data source [<a href="#B27-entropy-17-07680" class="html-bibr">27</a>] and agrees with those in the left panel. (Right, bottom) The Jaccard <math display="inline"> <msub> <mi>J</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </math> matrix measuring the resource overlap of species i and j. <math display="inline"> <mrow> <msub> <mi>J</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math> (dark blue) corresponds to no resource overlap, while <math display="inline"> <mrow> <msub> <mi>J</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math> (dark red) corresponds to complete resource overlap, in general only possible between two specialists with <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> that have the same resource. Full article ">Figure 4
(Top to bottom) 3-, 4-, 6-, 7- and 9-fold EM classification of the DISH unipartite pollinator-pollinator network. As the number of classes increase, we see both robustness, namely certain species being classified together, as well as the emergence of finer structure, as classes are split up (legend as in <a href="#entropy-17-07680-f002" class="html-fig">Figure 2</a>). Full article ">

Review

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291 KiB

Open AccessReview

Relative Entropy in Biological Systems

by John C. Baez and Blake S. Pollard

Entropy 2016, 18(2), 46; https://doi.org/10.3390/e18020046 - 2 Feb 2016

Cited by 46 | Viewed by 10435

Abstract

In this paper we review various information-theoretic characterizations of the approach to equilibrium in biological systems. The replicator equation, evolutionary game theory, Markov processes and chemical reaction networks all describe the dynamics of a population or probability distribution. Under suitable assumptions, the distribution [...] Read more.

In this paper we review various information-theoretic characterizations of the approach to equilibrium in biological systems. The replicator equation, evolutionary game theory, Markov processes and chemical reaction networks all describe the dynamics of a population or probability distribution. Under suitable assumptions, the distribution will approach an equilibrium with the passage of time. Relative entropy—that is, the Kullback–Leibler divergence, or various generalizations of this—provides a quantitative measure of how far from equilibrium the system is. We explain various theorems that give conditions under which relative entropy is nonincreasing. In biochemical applications these results can be seen as versions of the Second Law of Thermodynamics, stating that free energy can never increase with the passage of time. In ecological applications, they make precise the notion that a population gains information from its environment as it approaches equilibrium. Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

Other

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176 KiB

Open AccessCorrection

Correction: Wolpert, D.H. The Free Energy Requirements of Biological Organisms; Implications for Evolution. Entropy 2016, 18, 138

by David H. Wolpert

Entropy 2016, 18(6), 219; https://doi.org/10.3390/e18060219 - 2 Jun 2016

Cited by 3 | Viewed by 4110

Abstract

The following corrections should be made to the published paper [1]: [...] Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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