Stationary distribution of selforganized states and biological information generation.  
Jump to Full Text  
MedLine Citation:

PMID: 24281357 Owner: NLM Status: InDataReview 
Abstract/OtherAbstract:

Selforganization, where spontaneous orderings occur under driven conditions, is one of the hallmarks of biological systems. We consider a statistical mechanical treatment of the biased distribution of such organized states, which become favored as a result of their catalytic activity under chemical driving forces. A generalization of the equilibrium canonical distribution describes the stationary state, which can be used to model shifts in conformational ensembles sampled by an enzyme in working conditions. The basic idea is applied to the process of biological information generation from random sequences of heteropolymers, where unfavorable Shannon entropy is overcome by the catalytic activities of selected genes. The ordering process is demonstrated with the genetic distance to a genotype with high catalytic activity as an order parameter. The resulting free energy can have multiple minima, corresponding to disordered and organized phases with firstorder transitions between them. 
Authors:

Hyung Jun Woo 
Related Documents
:

23409287  ‘big data’ from shrinking pathogen populations. 24786517  Statistical modeling reveals the effect of absolute humidity on dengue in singapore. 25095277  Differential evolution with autoenhanced population diversity. 23654027  Measurement equivalence of seven selected items of posttraumatic growth between black a... 16237657  Bootstrap analysis of multivariate failure time data. 24221377  Selection of sires to reduce sampling variance in the estimates of heritability by half... 
Publication Detail:

Type: Journal Article Date: 20131125 
Journal Detail:

Title: Scientific reports Volume: 3 ISSN: 20452322 ISO Abbreviation: Sci Rep Publication Date: 2013 
Date Detail:

Created Date: 20131127 Completed Date:  Revised Date:  
Medline Journal Info:

Nlm Unique ID: 101563288 Medline TA: Sci Rep Country: England 
Other Details:

Languages: eng Pagination: 3329 Citation Subset: IM 
Affiliation:

Henry M. Jackson Foundation for the Advancement of Military Medicine, 2405 Whittier Drive, Frederick, Maryland, 21702, USA. 
Export Citation:

APA/MLA Format Download EndNote Download BibTex 
MeSH Terms  
Descriptor/Qualifier:

Full Text  
Journal Information Journal ID (nlmta): Sci Rep Journal ID (isoabbrev): Sci Rep ISSN: 20452322 Publisher: Nature Publishing Group 
Article Information Download PDF Copyright © 2013, Macmillan Publishers Limited. All rights reserved openaccess: Received Day: 04 Month: 09 Year: 2013 Accepted Day: 08 Month: 11 Year: 2013 Electronic publication date: Day: 25 Month: 11 Year: 2013 collection publication date: Year: 2013 Volume: 3Elocation ID: 3329 PubMed Id: 24281357 ID: 3839033 Publisher Item Identifier: srep03329 DOI: 10.1038/srep03329 
Stationary distribution of selforganized states and biological information generation  
Hyung Jun Wooa1  
1Henry M. Jackson Foundation for the Advancement of Military Medicine, 2405 Whittier Drive, Frederick, Maryland, 21702, USA 

ahgjun.woo@gmail.com 
Despite enormous progress in understanding the characteristics of life's building blocks and their interactions^{1}, many aspects of processes occurring in living organisms continue to pose challenges to physicsbased explanations. A major difficulty is in characterizing their organization and function, which tend to appear spontaneously under suitable conditions, in stark contrast to common experience with nonliving matter ruled by increasing disorder. The term selforganization has been used widely to describe these spontaneous appearances of highly ordered structures, not only within biological systems but also in higherlevel organizations including networks^{2}^{, 3}^{, 4}^{, 5}^{, 6}, and complex dynamical systems exhibiting phenomena such as disease progression^{7}^{, 8}^{, 9} and neural computation^{10}. Different directions of theoretical approaches include wideranging studies of driven systems showing selforganized criticality^{11}^{, 12} with implications to extinction dynamics^{13}, dynamical systems views with analogies to equilibrium phase transition^{14}, and concepts centered on autocatalysis, evolution, and selection^{15}. Further studies in this interdisciplinary field include models of genetic regulatory networks and cell differentiation^{16}, dynamic clustering in active media^{17}, and the study of Boolean network dynamics^{18}. However, one characteristic common to current approaches of studying selforganization is the lack of concrete connections to equilibrium statistical mechanics.
We focus in this paper on chemically driven systems and describe an approach extending the equilibrium statistical mechanical concepts to cover the stationary distribution of selforganized states. Our approach, which is based on a combination of equilibrium theory and enzyme kinetics, will allow us to distinguish selforganization from selfassembly, a related but distinct class of phenomena in which ordered structures are favored in equilibrium because of certain structural features present within the constituents. A familiar example is the formation of micelles and lipid bilayers stabilized by hydrophobic interactions^{19}, for which wellestablished and quantitative theories now exist^{20}.
Selforganization, in contrast, is a sustainable nonequilibrium process in which a system spontaneously increases and maintains its degree of ordering as a result of interactions and exchanges of matter with its surrounding. Typically the system is in thermal and barometric equilibrium with its surrounding, but is driven by supply and extraction of chemical species; the ‘food’ and ‘waste’ molecules. The point of view adopted in this paper is that the ordering occurs when the system has the potential to catalyze reactions involving these externally controlled species in the favored direction. We use a simple description of this effect to derive a biased distribution of system configurations away from equilibrium. The driving force increasingly favors organized states that would have negligible probabilities of occupation in equilibrium.
One immediate consequence of such effects amenable to current experimental investigations is the shift in conformational distributions of protein enzymes while under stationary working conditions compared to equilibrium. Recent developments in singlemolecule experimental techniques^{21}^{, 22} have uncovered many surprises challenging the traditional views on how enzymes operate, including the classical ‘induced fit’ mechanism, which assumed that an enzyme mostly inhabits a single conformation, only changing it as a result of substrate binding. Increasing body of experimental evidence suggests that many enzymes instead sample a wide range of conformational states without ligands, while a substrate binding event shifts the equilibrium into states optimal for catalytic activities (‘conformational selection’ mechanism)^{23}^{, 24}^{, 25}. Theoretical studies so far have focused mostly on the effects this conformational heterogeneity has on kinetic velocity^{25}^{, 26}. The theoretical consideration presented in this paper provides a simple statistical mechanical description of the nonequilibrium enzyme conformational distribution that can be used to interpret singlemolecule experiments.
As a second, more fundamental application, we address the issue of how modern proteins capable of exhibiting such ordering that defies entropic costs – e.g., via conformational changes leading to the activation of its enzymatic activity – or rather the biological information encoding their structure and function, could have arisen spontaneously. Viewed from the information theoretic perspective^{27}, the generation of such biological information is a selforganization process where random heteropolymers with maximum entropy were replaced by highly conserved genes. Considerations of this process can serve as a bridge between equilibrium statistical mechanics and current welldeveloped statistical approaches to modeling evolution^{28}, as well as highly successful recent developments in datadriven characterizations of biological genotypephenotype spaces^{29}^{, 30} and reconstruction of the evolutionary history of extant metabolic pathways^{31}.
There are two broad classes of approaches put forward to describe such an ordering of biochemical sequences. One highly influential perspective is to center on the properties of the minimal networks of autocatalytic species^{15}^{, 32}^{, 33}^{, 34}^{, 35}^{, 36}, where selforganization becomes possible when the degree of diversity of the autocatalytic set exceeds a critical value. A different approach, represented by the quasispecies theory^{37}^{, 38} and theoretical works based on it^{39}^{, 40}^{, 41}^{, 42}^{, 43}^{, 44}^{, 45}, takes the selfreplication population dynamics of sequences as a starting point. The orderdisorder transition of genetic information occurs as the mutation rate of the replication process crosses a threshold value.
Based on the general consideration of the stationary distribution under driven conditions, we consider in this paper the question of whether the ordering in sequence space can occur purely from chemical driving forces without the mechanism of competition based on selfreplication postulated in quasispecies theory. It is shown that there exists a firstorder transition from the phase characterized by random sequences to those dominated by the few active sequences irrespective of the specific population dynamics of heteropolymers. Our thermodynamic consideration thus connects the biochemical selforganization more directly to chemical driving forces and reveals its close correspondence to equilibrium phase transitions, complementing the autocatalytic kineticsbased and populationbased approaches.
We consider a system in thermal equilibrium with a reservoir (Fig. 1), characterized by a set of (coarsegrained) states n and corresponding (free) energy E_{n}. The equilibrium canonical distribution of finding the system in state n is (we measure energy and entropy in units of temperature and Boltzmann constant, respectively). An example of n is a variable indicating whether an enzyme is in one conformational state or another, defined for instance in terms of an angle or distance between certain subdomains within the protein. When driven by the reservoir, each state n has some degree of catalytic activity toward a reaction R → P imposed by the reservoir:
In the absence of driving forces from the reservoir, the set of states {n} satisfies the detailed balance condition, , which remains valid even when , because Eq. (3) does not couple n and m. This assumption could be violated if for instance the reaction scheme Eq. (1) is generalized such that S_{n} · R could turn into states m ≠ n, which may yield useful models for molecular motors^{47}^{, 48}^{, 49}. We will not consider such cases in this paper. In contrast to Eq. (2), the two main reaction steps in Eq. (1) do not satisfy detailed balance except in equilibrium.
We have , and may thus write
From Eq. (1), the stationary velocity v can be written as
A simplest special case is where there are only two states, n = D, O, each corresponding to disordered and organized states (Fig. 1), and ΔE = E_{O} − E_{D} > 0. From Eq. (8), if and c_{p} = 0, the relative stability of D over O is reversed when . Typical K_{m} values of enzymes range from μM to mM ranges^{1}. For an enzyme with K_{m} = 1 μM, for instance, for ΔE = 10. It is worth noting that in contrast to selfassembly, the stabilization of states with ΔE > 0 arises strictly from the chemical driving force of the reservoir. If the matter flow is cut off, the system would quickly revert to D: the organized structure in the system ‘dies’.
For the two state model, Eq. (11) becomes
More general cases can be illustrated further by a simple toy model of an enzyme with a continuous angular degree of freedom θ. This angle represents the degree of closing of the binding pocket at the center. As θ becomes smaller, the catalytic activity increases linearly, while the thermal stability decreases:
Virtually all selforganized structures in biological systems are based on biopolymers, including proteins and nucleic acids (RNAs and DNAs), which carry biological information that are copied over generations with mutations. A satisfactory theory of selforganization therefore must address how such biological information could have been generated. We show below that the chemical driving force imposed by the reservoir induces an orderdisorder transition in sequence space, where the stationary distribution of genotypes becomes dominated by sequences with catalytic activity.
We consider a chemically driven environment where nucleotide sequences of a fixed length l are continually synthesized and degraded, e.g., against a solid support. Each nucleotide at different sites on the chain can contain one of four possible bases, b = a, g, c, u, with the total number of all possible sequences s_{n} = {b_{1}, …, b_{l}}, Ω = 4^{l}. Without additional driving forces other than the chain synthesis, the resulting sequences would be mostly random. This pool of random sequences corresponds to the system in the disordered D state in Fig. 1. The stationary probability P_{n} for an RNA chain randomly picked from the population to have sequence s_{n} is , where E_{n} describes the intrinsic relative stability of each sequence.
If the reservoir exerts a driving force F = μ_{r} − μ_{p} > 0 for a reaction R → P, Eq. (1) describes the catalysis with S_{n} referring to a chain with sequence n: RNA chains with suitable sequences can fold and catalyze reactions, as does the catalytic core of modern ribosomes carrying out the synthesis of proteins during translation^{51}^{, 52}. Equation (8) gives the biased energy under the driven condition, where we assume E_{n} = 0 for simplicity. We note that z_{n} given by Eq. (5) in this case represents the enzymatic activity of the sequence n, or its fitness: the generalized free energy is more negative for sequences with higher fitness. Equation (7) therefore implies that as the system reaches the stationary state under the influence of reservoir driving force, the distribution would become peaked around sequences with low energy, or high fitness. The analogy of the evolution of populations toward genotypes with higher fitness – the climbing of peaks on the fitness landscape – to the minimization of energy in equilibrium systems has been observed before, notably by Kauffman^{15} (but also note the proposal that fitness landscapes are intrinsically dynamic quantities^{53}). Here, we see this correspondence more directly from thermodynamic considerations.
This identification of the fitness of genotypes as their catalytic activity contains its common definition in evolutionary theories – the replication rate – as a special case where R → P is the selfreplication reaction. The orderdisorder transition we describe below, on the other hand, does not rely on the assumption of the existence of selfreplication machinery, and may provide an understanding of how the large variety of genes found in modern genomes coding for enzymes with many different functions could have originated.
To describe the biased population of sequences under driven conditions in more detail, the fitness (or energy) landscape needs to be specified. Physically, we only need to be convinced of the existence of sequences with high catalytic activities toward the reaction imposed by the reservoir. The fitness landscape describes the dependence of fitness values as we depart from these genotypes in sequence space. The theory of selforganization allows for a quantitative description of the dominance of highly fit genotypes with a discontinuous firstorder transition.
For concreteness, we adopt the class of landscapes widely studied in applications of the quasispecies theory^{28}^{, 37}^{, 39}^{, 41}^{, 42}^{, 54}, where the fitness is given as a function of Hamming distance to the master sequence, which becomes the natural order parameter. It is worth emphasizing, however, that in contrast to the error catastrophe transition in the quasispecies theory, the orderdisorder transition we derive below is thermodynamic in origin independent of specific population dynamics, and is applicable to any reaction for which a sufficiently strong enzymatic genotype exists in the sequence space.
The distance h_{nm} between two sequences n and m is the total number of nucleotide positions at which the base identities differ. For lmers, 0 ≤ h ≤ l. The probability P_{h} of sequences with distance h is
In Fig. 3(a), a model fitness landscape
The qualitative features of G_{h} for different driving forces resembles the orderdisorder transitions observed in equilibrium fluids, where a condensed phase can coexist with the disordered phase. In the corresponding phase diagram shown in Fig. 3(b) as a function of distance h and driving force ζ, the coexistence region between the disordered D phase and the organized O phase shrinks as the fitness peak width ξ increases and ζ decreases, vanishing at a critical point where the DO transition becomes continuous. The D phase is characterized by random sequences, while in the O phase, the sequence distribution is peaked at a short distance from the catalytically active master sequence. The position of the minimum in G_{h} in the O phase in Fig. 3(a) corresponds to the most likely distance value h expected within this population.
The transition from the disordered to organized phases is accompanied by the dominance of catalytically active states leading to a bias in distribution P_{n}, under which sequences carry information encoding enzymes. The amount of this information generated is quantified by the reduction in Gibbs entropy associated with P_{n}, known as the information content (per site)^{27}^{, 55}^{, 56},
Equation (1) can be easily generalized to cases where there are more than one externally imposed reactions, which would lead to the coexistence of multiple genes. We consider the case of two reactions (e.g., RNA elongation and another reaction such as ATP hydrolysis). The order parameter is a vector h = (h_{1}, h_{2}) with two components specifying distances to the master sequences s_{1} and s_{2}. It can then be shown that Eq. (8) becomes
The calculation of Ω_{h} involves counting the number of sequences with given distances h to the master sequences. In Fig. 5, the two master sequences s_{1} and s_{2} are depicted with their sites grouped into two sections, I and II, where nucleotides are different and identical between the sequences, respectively. The length of section I is h_{12} = d. To count the number of sequences s_{n} with distances h_{1} and h_{2} to s_{1} and s_{2}, we start with s_{1} and first mutate a subset of length m from section II, such that h = (m, d + m). The number of ways of doing this is because there are three nucleotides different from each site in section II. We then mutate k sites from section I into the corresponding nucleotides of s_{2}, which results in h = (m + k, d + m − k). The number of ways of doing this step is without any nucleotide multiplicity because the target sequence is fixed. By choosing k = d + m − h_{2}, the distance h_{2} to s_{2} is achieved. Finally, we choose p additional sites from section I and mutate into nucleotides distinct from both s_{1} and s_{2}, after which h = (m + k + p, d + m − k). The number of ways for this third step is because there are two nucleotides that can be chosen for each site. Taking p = h_{1} − m − k = h_{1} + h_{2} − d − 2m, we achieve the distance h_{1} to s_{1}. The total number of sequences is then given by
Figure 6 shows the free energy as a function of h for equilibrium (ζ_{i} = 0) and strongly driven (ζ_{1} = ζ_{2} = 10^{4}) cases. The landscape is bounded by values of h for which Ω_{h} = 0. The allowed region can be deduced by requiring m_{0} ≤ m_{1} in Eqs. (21): 0 ≤ h_{i} ≤ l, h_{1} ≤ h_{2} + d, h_{2} ≤ h_{1} + d, and h_{1} + h_{2} ≥ d. The single minimum in Fig. 6(a) at h = (8, 8) corresponds to the disordered D phase, which is the only phase in equilibrium. Under strong driving forces, in contrast, up to two additional minima develop near the h_{2} and h_{1}axes [Fig. 6(b)], which correspond to organized O_{1} and O_{2} phases dominated by sequences close to s_{1} and s_{2}, respectively. The global minimum switches from D to O_{i} (or both when ζ_{1} = ζ_{2}) with increasing ζ_{i}.
The phase diagram in Fig. 7 shows such stability changes within the ζ_{i}space, where the three phases are separated by boundaries on which neighboring phases can coexist. There is a triple point where D, O_{1} and O_{2} phases can all coexist. As suggested by the singlegene case in Fig. 3(b), increasing the fitness peak width parameter ξ shifts the DO_{i} boundaries toward smaller ζ_{i} values. The boundaries disappear at a critical point. The symmetry 1 ↔ 2 in Figs. 6 and 7 is a result of our choice of the same fitness landscape, Eq. (17), for the two genes, and would be broken in more general cases.
In this paper, we presented a statistical mechanical description of biological selforganization under chemically driven conditions: the system possesses a small subset of coarsegrained states which can catalyze the reactions imposed externally by the reservoir. The entropic cost of observing these organized states is overcome when the driving forces are sufficiently strong, leading to biased stationary distributions dominated by such organized states, or phases.
As a major application, we focused on the appearance of biological information encoded into sequences of nucleotide strands. Chemical driving forces lead to one or multiple peaks in sequence space, centered on sequences that can catalyze the externally imposed reactions. In this viewpoint, genes encoding enzymes spontaneously appear and dominate the nucleotide sequence populations if the driving forces are sufficiently strong to overcome the Shannon entropy costs. Our analyses demonstrate that this transition into selforganization in sequence space has much in common with equilibrium phase transitions. Although the global stability of phases is dictated by the phase diagram (Fig. 7) as in equilibrium, metastable phases (and genes) can still be present together with the dominant phase outside the coexistence region, with the relative population of different sequences given by Eq. (7).
The phase transition observed here can occur irrespective of how interconversion between sequences actually takes place (random synthesis on solid support or replication of existing chains). The nature of sequence evolution, however, would affect the dynamics of ordering, which was not considered here. The establishment of stationary distribution, Eq. (7), requires sufficient exploration of all sequences via Eq. (2), fastest with random synthesis but still taking relaxation times that grow exponentially with l. Highfidelity replications would slow down this relaxation, while allowing for the preservation of information already discovered.
The organized phases in Figs. 6 and 7 consist of groups of heteropolymers independently acting as enzymes. One of these groups can be polymerases catalyzing the synthesis of polymers. An aspect of evolutionary transitions we may presume to have occurred, in particular, is that of the emergence of ‘selfishness’, or the ability of the polymerase gene to limit its action to its own replication only, excluding others. This assumption is one of the starting points of the quasispecies theory^{37}. The selfishness is also likely to be closely related to the conjoining of genes into genomes. It will be of interest to see how we may understand the evolution of selfishness within the statistical mechanical perspective.
The viewpoint we adopted for biological selforganization – the stabilization of structures capable of catalyzing reactions imposed by chemical driving forces – may also have relevance to the question of how one may usefully define living organisms. RuizMirazo et al.^{57} emphasized two main elements in such a definition: autonomy and openended evolution. The former is a subset of selforganization capable of autoregulation, while the latter requires the establishment of a division of labor between recordkeeping (DNA) and expression (proteins) of biological information. The perspective adopted and elaborated in this paper shows how thermodynamic driving forces both constrain and enable selforganization, which may prove useful in understanding higherlevel structures.
The statistical mechanical expression for stationary states and its partition function, Eq. (9), can form a basis for calculating properties of systems with more complex features than assumed here. In particular, one may have multiple reactions coupled to each other, a common situation in biochemical systems, which would lead to an extension of Eq. (19) to a coupled ‘hamiltonian’. The calculation of partition function would be akin to that for interacting systems in equilibrium such as the Ising model. Such an extension would also allow one to consider fairly large systems in which the system components catalyze the formation of one another, forming an autocatalytic network^{15}^{, 32}^{, 33}. In such systems, the thermodynamic phase transitions studied here may therefore precede and combine with the transition to selfsustained autocatalytic organizations.
For the toy model defined by Eq. (13), after replacing discrete sums with integrals and adding a field f, Eq. (9) reads
The phase diagram in Fig. 3(b) was obtained by varying the width parameter ξ of the landscape (17), and locating the values of the driving force ζ for which the two phases – organized (O) and disordered (D) – have the same free energy. The coexistence values of ζ decreases with increasing ξ from top to bottom.
The validity of Eq. (20) for the total number of sequences with given distances to two master sequences was verified by enumerating all genotypes for small l and counting the number of sequences for each set of possible distance values.
H.J.W. performed the research and wrote the manuscript.
Notes
The author declares no competing financial interests.
Berg J. M.,, Tymoczko J. L., & Stryer L.. Biochemistry (Freeman, New York, Year: 2002).  
Newman M. E. J.,Networks: An Introduction (Oxford, New York, Year: 2010).  
Barabási A.L., & Albert R.,Emergence of scaling in random networks. Science286, 509–512 (Year: 1999).10521342  
Ahn Y.Y.,, Ahnert S. E.,, Bagrow J. P., & Barabási A.L.,Flavor network and the principles of food pairing. Sci. Rep.1, 196 (Year: 2011).22355711  
Bagrow J. P.,Communities and bottlenecks: trees and treelike networks have high modularity. Phys. Rev. E85, 066118 (Year: 2012).  
ColomerdeSimón P.,, Serrano M. Á.,, Beiró M. G.,, AlvarezHamelin J. I., & Boguñá M.,Deciphering the global organization of clustering in real complex networks. Sci. Rep.3, 2517 (Year: 2013).23982757  
Vespignani A.,Modelling dynamical processes in complex sociotechnical systems. Nat. Phys.8, 32–39 (Year: 2012).  
Meloni S.,et al.Modeling human mobility responses to the largescale spreading of infectious diseases. Sci. Rep.1, 62 (Year: 2011).22355581  
Boguñá M.,, Castellano C., & PastorSatorras R.,Nature of the epidemic threshold for the susceptibleinfectedsusceptible dynamics in networks. Phys. Rev. Lett.111, 068701 (Year: 2013).23971619  
Hopfield J. J.,Physics, computation, and why biology looks so different. J. Theor. Biol.171, 53–60 (Year: 1994).  
Bak P.,, Tang C., & Wiesenfeld K.,Selforganized criticality: an explanation of the 1/f noise. Phys. Rev. Lett.59, 381–384 (Year: 1987).10035754  
Drossel B.,Complex scaling behavior of nonconserved selforganized critical systems. Phys. Rev. Lett.89, 238701 (Year: 2002).12485047  
Newman M. E. J., & Palmer R. G.,Modeling Extinction (Oxford, New York, Year: 2003).  
Haken H.,Synergetics: An Introduction (Springer, Berlin, Year: 1983).  
Kauffman S. A.,The Origins of Order: SelfOrganization and Selction in Evolution (Oxford, New York, Year: 1993).  
Hanel R.,, Pöchacker M.,, Schölling M., & Thurner S.,A selforganized model for celldifferentiation based on variations of molecular decay rates. PLOS ONE7, e36679 (Year: 2012).22693554  
Theurkauff I.,, CottinBizonne C.,, Palacci J.,, Ybert C., & Bocquet L.,Dynamic clustering in active colloidal suspensions with chemical signaling. Phys. Rev. Lett.108, 268303 (Year: 2012).23005020  
Gehrmann E., & Drossel B.,Boolean versus continuous dynamics on simple twogene modules. Phys. Rev. E82, 046120 (Year: 2010).  
Safran S. A.,Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (AddisonWesley, New York, Year: 1994).  
Chandler D.,Interfaces and the driving force of hydrophobic assembly. Nature437, 640–647 (Year: 2005).16193038  
Min W.,et al.Flucuating enzymes: lessons from singlemolecule studies. Acc. Chem. Res.38, 923–931 (Year: 2005).16359164  
Kou S. C.,, Cherayil B. J.,, Min W.,, English B. P., & Xie X. S.,Singlemolecule MichaelisMenten equations. J. Phys. Chem. B109, 19068–19081 (Year: 2005).16853459  
Bohr D. D.,, Nussinov R., & Wright P. E.,The role of dynamic conformational ensembles in biomolecular recognition. Nat. Chem. Biol.5, 789–796 (Year: 2009).19841628  
Ma B., & Nussinov R.,Enzyme dynamics point to stepwise conformational selection in catalysis. Curr. Opin. Chem. Biol.14, 652–659 (Year: 2010).20822947  
Weikl T. R., & von Deuster C.,Selectedfit versus inducedfit protein binding: kinetic differences and mutational analysis. Proteins75, 104–110 (Year: 2009).18798570  
Min W.,, Xie X. S., & Bagchi B.,Role of conformational dynamics in kinetics of an enzymatic cycle in a nonequilibrium steady state. J. Chem. Phys.131, 065104 (Year: 2009).19691414  
Shannon C. E., & Weaver W.,The Mathematical Theory of Communication (Univ. of Illinois Press, Urbana, Year: 1949).  
Drossel B.,Biological evolution and statistical physics. Adv. Phys.50, 209–295 (Year: 2001).  
Ferrada E., & Wagner A.,Evolutionary innovations and the organization of protein functions in genotype space. PLOS ONE5, e14172 (Year: 2010).21152394  
Ferrada A., & Wagner A.,Protein robustness promotes evolutionary innovations on large evolutionary timescales. Proc. Roy. Soc. BBiol. Sci.275, 1595–1602 (Year: 2008).  
Braakman R., & Smith E.,The emergence and early evolution of biological carbonfixation. PLOS Comput. Biol.8, e1002455 (Year: 2012).22536150  
Farmer J. D.,, Kauffman S. A., & Packard N. H.,Autocatalytic replication of polymers. Physica D22, 50–67 (Year: 1986).  
Kauffman S. A.,Autocatalytic sets of proteins. J. Theor. Biol.119, 1–24 (Year: 1986).3713221  
Hanel R.,, Kauffman S. A., & Thurner S.,Phase transition in random catalytic networks. Phys. Rev. E72, 036117 (Year: 2005).  
Fontana W., & Buss L. W.,“The arrival of the fittest”: toward a theory of biological organization. B. Math. Biol.56, 1–64 (Year: 1994).  
Dittrich P., & di Fenizio P. S.,Chemical organization theory. B. Math. Biol.69, 1199–1231 (Year: 2007).  
Eigen M.,Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften58, 465–523 (Year: 1971).  
Swetina J., & Schuster P.,Selfreplication with errors. A model for polynucleotide replication. Biophys. Chem.16, 329–345 (Year: 1982).7159681  
Tannenbaum E., & Shakhnovich E. I.,Semiconservative replication, genetic repair, and manygened genomes: extending the quasispecies paradigm to living systems. Phys. Life Rev.2, 290–317 (Year: 2005).  
Altmeyer S., & McCaskill J. S.,Error threshold for spatially resolved evolution in the quasispecies model. Phys. Rev. Lett.86, 5819–5822 (Year: 2001).11415366  
Saakian D. B., & Hu C.K.,Exact solution of the Eigen model with general fitness functions and degradation rates. Proc. Natl. Acad. Sci. USA103, 4935–4939 (Year: 2006).16549804  
Saakian D. B.,, Muñoz E.,, Hu C.K., & Deem M. W.,Quasispecies theory for multiplepeak fitness landscapes. Phys. Rev. E73, 041913 (Year: 2006).  
Muñoz E.,, Park J.M., & Deem M. W.,Quasispecies theory for horizontal gene transfer and recombination. Phys. Rev. E78, 061921 (Year: 2008).  
Park J.M.,, Muñoz E., & Deem M. W.,Quasispecies theory for finite populations. Phys. Rev. E81, 011902 (Year: 2010).  
Woo H.J., & Wallqvist A.,Nonequilibrium phase transitions associated with DNA replication. Phys. Rev. Lett.106, 060601 (Year: 2011).21405451  
Csermely P.,, Palotai R., & Nussinov R.,Induced fit, conformational selection and independent dynamic segments: an extended view of binding events. Trends. Bichem. Sci.35, 539–546 (Year: 2010).  
Reimann P.,Brownian motors: noisy transport far from equilibrium. Phys. Rep.361, 57–265 (Year: 2002).  
Woo H.J.,Analytical theory of the nonequilibrium spatial distribution of RNA polymerase translocations. Phys. Rev. E74, 011907 (Year: 2006).  
Woo H.J.,Relaxation dynamics near nonequilibrium stationary states in Brownian ratchets. Phys. Rev. E79, 021101 (Year: 2009).  
Callen H. B.,Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, Year: 1985).  
Cech T. R.,Structural biology. The ribosome is a ribozyme. Science289, 878–879 (Year: 2000).10960319  
Joyce G. F.,The antiquity of RNAbased evolution. Nature418, 214–221 (Year: 2002).12110897  
Klimek P.,, Thurner S., & Hanel R.,Evolutionary dynamics from a variational principle. Phys. Rev. E82, 011901 (Year: 2010).  
Itan E., & Tannenbaum E.,Semiconservative quasispecies equations for polysomic genomes: the general case. Phys. Rev. E81, 061915 (Year: 2010).  
Schneider T. D.,, Stormo G. D.,, Gold L., & Ehrenfeucht A.,Information content of binding sites on nucleotide sequences. J. Mol. Biol.188, 415–431 (Year: 1986).3525846  
Adami C.,Information theory in molecular biology. Phys. Life Rev.1, 3–22 (Year: 2004).  
RuizMirazo K.,, Peretó J., & Moreno A.,A universal definition of life: autonomy and openended evolution. Origins Life Evol. B.34, 323–346 (Year: 2004). 
Article Categories:

Previous Document: Amplitude spectrum EEG signal evidence for the dissociation of motor and perceptual spatial working ...
Next Document: Application of Raman spectroscopy for visualizing biochemical changes during peripheral nerve injury...