On Simplified Global Nonlinear Function for Fitness Landscape: A Case Study of Inverse Protein Folding

The construction of fitness landscape has broad implication in understanding molecular evolution, cellular epigenetic state, and protein structures. We studied the problem of constructing fitness landscape of inverse protein folding or protein design, with the aim to generate amino acid sequences that would fold into an a priori determined structural fold which would enable engineering novel or enhanced biochemistry. For this task, an effective fitness function should allow identification of correct sequences that would fold into the desired structure. In this study, we showed that nonlinear fitness function for protein design can be constructed using a rectangular kernel with a basis set of proteins and decoys chosen a priori. The full landscape for a large number of protein folds can be captured using only 480 native proteins and 3,200 non-protein decoys via a finite Newton method. A blind test of a simplified version of fitness function for sequence design was carried out to discriminate simultaneously 428 native sequences not homologous to any training proteins from 11 million challenging protein-like decoys. This simplified function correctly classified 408 native sequences (20 misclassifications, 95% correct rate), which outperforms several other statistical linear scoring function and optimized linear function. Our results further suggested that for the task of global sequence design of 428 selected proteins, the search space of protein shape and sequence can be effectively parametrized with just about 3,680 carefully chosen basis set of proteins and decoys, and we showed in addition that the overall landscape is not overly sensitive to the specific choice of this set. Our results can be generalized to construct other types of fitness landscape.     

Comparative statics of games between relatives

According to Hamilton's theory of kin selection, species tend to evolve behavior such that each organism appears to be attempting to maximize its inclusive fitness. In particular, two neighbors are likely to help each other if the cost of doing so is less than the benefit multiplied by r, their coefficient of relatedness. Since the latter is less than unity, mutual altruism benefits both neighbors. However, is it theoretically possible that acting so as to maximize the inclusive, rather than personal, fitness may harm both parties. This may occur in strategic symmetric pairwise interactions (more specifically, nxn games), in which the outcome depends on both sides' actions. In this case, the equilibrium outcome may be less favorable to the interactants' personal fitness than if each of them acted so as to maximize the latter. This paper shows, however, that such negative effect of relatedness on fitness is incompatible with evolutionary stability. If the symmetric equilibrium strategies are evolutionarily stable, a higher coefficient of relatedness can only entail higher personal fitness for the two neighbors. This suggests that negative comparative statics as above are not likely to occur in nature.     

Optimal competitive hopfield network with stochastic dynamics for maximum cut problem

In this paper, introducing stochastic dynamics into an optimal competitive Hopfield network model (OCHOM), we propose a new algorithm that permits temporary energy increases which helps the OCHOM escape from local minima. The goal of the maximum cut problem, which is an NP-complete problem, is to partition the node set of an undirected graph into two parts in order to maximize the cardinality of the set of edges cut by the partition. The problem has many important applications including the design of VLSI circuits and design of communication networks. Recently, Galán-Marín et al. proposed the OCHOM, which can guarantee convergence to a global/local minimum of the energy function, and performs better than the other competitive neural approaches. However, the OCHOM has no mechanism to escape from local minima. The proposed algorithm introduces stochastic dynamics which helps the OCHOM escape from local minima, and it is applied to the maximum cut problem. A number of instances have been simulated to verify the proposed algorithm.     

Developing optimal non-linear scoring function for protein design

Motivation. Protein design aims to identify sequences compatible with a given protein fold but incompatible to any alternative folds. To select the correct sequences and to guide the search process, a design scoring function is critically important. Such a scoring function should be able to characterize the global fitness landscape of many proteins simultaneously. RESULTS: To find optimal design scoring functions, we introduce two geometric views and propose a formulation using a mixture of non-linear Gaussian kernel functions. We aim to solve a simplified protein sequence design problem. Our goal is to distinguish each native sequence for a major portion of representative protein structures from a large number of alternative decoy sequences, each a fragment from proteins of different folds. Our scoring function discriminates perfectly a set of 440 native proteins from 14 million sequence decoys. We show that no linear scoring function can succeed in this task. In a blind test of unrelated proteins, our scoring function misclassfies only 13 native proteins out of 194. This compares favorably with about three-four times more misclassifications when optimal linear functions reported in the literature are used. We also discuss how to develop protein folding scoring function.     

A simple procedure to weight empirical potentials in a fitness function so as to optimize its performance in ab initio protein-folding problem

In an approach to the protein folding problem by a Genetic Algorithm, the fitness function plays a critical role. Empirical potentials are generally used to build the fitness function, and they must be weighted to obtain a valuable one. The weights are generally found by the comparison with a set of misfolded structures (decoys), but a dependence of the obtained fitness generally arises on the used decoys. Here we describe a general procedure to find out, from a given set of potentials, their better linear combination that could either identify the wild structure or prove their powerlessness. We use topological considerations over the hyperspace of the potentials, and a multiple linear inequalities solver. The iterated method flows through the following steps: it determines a direction in the hyperspace of the potentials, which identifies the native structure as a vertex among a set of misfolded decoys. A multiple linear inequalities solver obtains the direction. A Genetic Algorithm, tailored to the specific problem, uses the fitness function defined by this direction and generally reaches a new structure better than the experimental one, which is added to the ensemble. The decoys so generated are not dependent on a deterministic criterion. This iterative procedure can be stopped either by identifying an effective fitness function or by proving the impossibility of its achievement. In order to test the method under the hardest conditions, we choose numerous and heterogeneous quantities as components of the fitness function. This method could be a useful tool for the scientific community in order to test any fitness proposed and to recognize the most important components on which it is built.     

Inclusive fitness and the sociobiology of the genome

Inclusive fitness theory provides conditions for the evolutionary success of a gene. These conditions ensure that the gene is selfish in the sense of Dawkins (The selfish gene, Oxford University Press, Oxford, 1976): genes do not and cannot sacrifice their own fitness on behalf of the reproductive population. Therefore, while natural selection explains the appearance of design in the living world (Dawkins in The blind watchmaker: why the evidence of evolution reveals a universe without design, W. W. Norton, New York, 1996), inclusive fitness theory does not explain how. Indeed, Hamilton’s rule is equally compatible with the evolutionary success of prosocial altruistic genes and antisocial predatory genes, whereas only the former, which account for the appearance of design, predominate in successful organisms. Inclusive fitness theory, however, permits a formulation of the central problem of sociobiology in a particularly poignant form: how do interactions among loci induce utterly selfish genes to collaborate, or to predispose their carriers to collaborate, in promoting the fitness of their carriers? Inclusive fitness theory, because it abstracts from synergistic interactions among loci, does not answer this question. Fitness-enhancing collaboration among loci in the genome of a reproductive population requires suppressing alleles that decrease, and promoting alleles that increase the fitness of its carriers. Suppression and promotion are effected by regulatory networks of genes, each of which is itself utterly selfish. This implies that genes, and a fortiori individuals in a social species, do not maximize inclusive fitness but rather interact strategically in complex ways. It is the task of sociobiology to model these complex interactions.     

The Cell Collective: Toward an open and collaborative approach to systems biology

Background

The construction of customized nucleic acid sequences allows us to have greater flexibility in gene design for recombinant protein expression. Among the various parameters considered for such DNA sequence design, individual codon usage (ICU) has been implicated as one of the most crucial factors affecting mRNA translational efficiency. However, previous works have also reported the significant influence of codon pair usage, also known as codon context (CC), on the level of protein expression.

Results

In this study, we have developed novel computational procedures for evaluating the relative importance of optimizing ICU and CC for enhancing protein expression. By formulating appropriate mathematical expressions to quantify the ICU and CC fitness of a coding sequence, optimization procedures based on genetic algorithm were employed to maximize its ICU and/or CC fitness. Surprisingly, the in silico validation of the resultant optimized DNA sequences for Escherichia coli, Lactococcus lactis, Pichia pastoris and Saccharomyces cerevisiae suggests that CC is a more relevant design criterion than the commonly considered ICU.

Conclusions

The proposed CC optimization framework can complement and enhance the capabilities of current gene design tools, with potential applications to heterologous protein production and even vaccine development in synthetic biotechnology.     

On the status of mean fitness maximization as a governing principle in evolution

Mean fitness is not always maximized under natural selection. In particular, in two locus models, recombination and epistasis may combine to prevent the operation of a maximizing priciple for mean fitness. If inversion phenomena are considered, however, there exist fully polymorphic equilibria which maximize the mean fitness and moreover the initial progress of an inversion can proceed if and only if it gives rise to an increase in mean fitness. While not applicable to all models, the principle of mean fitness maximization is still useful heuristically.     

Explaining the explosion: modelling hybrid invasions

The emergence of hybrids between native and introduced species is an increasingly widespread problem which can alter entire ecosystems. We present a general model for the hybridization of two plant species to investigate the conditions under which hybrid invasions can occur, and the ecological and genetic consequences of such hybridizations. We find that parental compatibility and fecundity are important determinants of whether (and at what rate) hybrid genotypes emerge. Enhanced hybrid fitness traits affect both the population's genetic structure and total rate of increase, with rapid selection for the fittest genotype. Conversely, if different genotypes maximize different life-history characteristics, the ensuing population can be genetically very variable. The model provides a novel approach to evaluate the contributions of population dynamic and genetic processes in the study of hybrid invasions.     

Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits

Summary We present models of adaptive change in continuous traits for the following situations: (1) adaptation of a single trait within a single population in which the fitness of a given individual depends on the population's mean trait value as well as its own trait value; (2) adaptation of two (or more) traits within a single population; (3) adaptation in two or more interacting species. We analyse a dynamic model of these adaptive scenarios in which the rate of change of the mean trait value is an increasing function of the fitness gradient (i.e. the rate of increase of individual fitness with the individual's trait value). Such models have been employed in evolutionary game theory and are often appropriate both for the evolution of quantitative genetic traits and for the behavioural adjustment of phenotypically plastic traits. The dynamics of the adaptation of several different ecologically important traits can result in characters that minimize individual fitness and can preclude evolution towards characters that maximize individual fitness. We discuss biological circumstances that are likely to produce such adaptive failures for situations involving foraging, predator avoidance, competition and coevolution. The results argue for greater attention to dynamical stability in models of the evolution of continuous traits.     

Bayes optimal informer sets for early-stage drug discovery

An important experimental design problem in early-stage drug discovery is how to prioritize available compounds for testing when very little is known about the target protein. Informer-based ranking (IBR) methods address the prioritization problem when the compounds have provided bioactivity data on other potentially relevant targets. An IBR method selects an informer set of compounds, and then prioritizes the remaining compounds on the basis of new bioactivity experiments performed with the informer set on the target. We formalize the problem as a two-stage decision problem and introduce the Bayes Optimal Informer SEt (BOISE) method for its solution. BOISE leverages a flexible model of the initial bioactivity data, a relevant loss function, and effective computational schemes to resolve the two-step design problem. We evaluate BOISE and compare it to other IBR strategies in two retrospective studies, one on protein-kinase inhibition and the other on anticancer drug sensitivity. In both empirical settings BOISE exhibits better predictive performance than available methods. It also behaves well with missing data, where methods that use matrix completion show worse predictive performance.     

Natural decision theory: A general formalism for the analysis of evolved characteristics

Classical Decision Theory, a mature and highly developed theory of rational choice, can be applied within evolutionary biology to the question of what traits an organism ought “rationally” to adopt, given that it wants to maximize its fitness. In this way the powerful formalism of decision theory can be brought to bear on the problem of how to predict which characters will be favored by natural selection, or to explain why certain characters have been so favored.Under some circumstances the classical theory of decision can be applied as it stands to an evolutionary problem simply by substituting an appropriate measure of biological fitness for the decision-theoretic concept of “utility”. Under other circumstances, however, it is necessary to extend the classical rules of decision in certain new directions. The result is a family of decision calculi of which the classical is only one. The name “Natural Decision Theory” is proposed for this extended class of biologically relevant decision methods.The decision tree method of diagramming an evolutionary decision situation is illustrated for the classical and three non-classical decision criteria, and is suggested as a potential means of gaining new insights into evolutionary forces.     

The linear interaction energy method for the prediction of protein stability changes upon mutation

The coupling of protein energetics and sequence changes is a critical aspect of computational protein design, as well as for the understanding of protein evolution, human disease, and drug resistance. To study the molecular basis for this coupling, computational tools must be sufficiently accurate and computationally inexpensive enough to handle large amounts of sequence data. We have developed a computational approach based on the linear interaction energy (LIE) approximation to predict the changes in the free-energy of the native state induced by a single mutation. This approach was applied to a set of 822 mutations in 10 proteins which resulted in an average unsigned error of 0.82 kcal/mol and a correlation coefficient of 0.72 between the calculated and experimental ΔΔG values. The method is able to accurately identify destabilizing hot spot mutations; however, it has difficulty in distinguishing between stabilizing and destabilizing mutations because of the distribution of stability changes for the set of mutations used to parameterize the model. In addition, the model also performs quite well in initial tests on a small set of double mutations. On the basis of these promising results, we can begin to examine the relationship between protein stability and fitness, correlated mutations, and drug resistance.     

Choosing fitness-enhancing innovations can be detrimental under fluctuating environments

The ability to predict the consequences of one's behavior in a particular environment is a mechanism for adaptation. In the absence of any cost to this activity, we might expect agents to choose behaviors that maximize their fitness, an example of directed innovation. This is in contrast to blind mutation, where the probability of becoming a new genotype is independent of the fitness of the new genotypes. Here, we show that under environments punctuated by rapid reversals, a system with both genetic and cultural inheritance should not always maximize fitness through directed innovation. This is because populations highly accurate at selecting the fittest innovations tend to over-fit the environment during its stable phase, to the point that a rapid environmental reversal can cause extinction. A less accurate population, on the other hand, can track long term trends in environmental change, keeping closer to the time-average of the environment. We use both analytical and agent-based models to explore when this mechanism is expected to occur.     

Optimal flight initiation distance

Decisions regarding flight initiation distance have received scant theoretical attention. A graphical model by Ydenberg and Dill (1986. The economics of fleeing from predators. Adv. Stud. Behav. 16, 229-249) that has guided research for the past 20 years specifies when escape begins. In the model, a prey detects a predator, monitors its approach until costs of escape and of remaining are equal, and then flees. The distance between predator and prey when escape is initiated (approach distance = flight initiation distance) occurs where decreasing cost of remaining and increasing cost of fleeing intersect. We argue that prey fleeing as predicted cannot maximize fitness because the best prey can do is break even during an encounter. We develop two optimality models, one applying when all expected future contribution to fitness (residual reproductive value) is lost if the prey dies, the other when any fitness gained (increase in expected RRV) during the encounter is retained after death. Both models predict optimal flight initiation distance from initial expected fitness, benefits obtainable during encounters, costs of escaping, and probability of being killed. Predictions match extensively verified predictions of Ydenberg and Dill's (1986) model. Our main conclusion is that optimality models are preferable to break-even models because they permit fitness maximization, offer many new testable predictions, and allow assessment of prey decisions in many naturally occurring situations through modification of benefit, escape cost, and risk functions.     

Social dynamics of mammals: Reproductive success, kinship and individual fitness

Apparent altruism, in which an individual seemingly decreases its evolutionary fitness by assisting others, can confer benefits if the individual assists kin. Thus, an animal can increase its total or inclusive fitness by producing offspring (direct fitness) and/or helping kin to reproduce (indirect fitness). Although kin selection has been suggested as the mechanism underlying the formation of mammalian societies, many species act as if they attempt to maximize the direct fitness component of their inclusive fitness.     

Stickbreaking: a novel fitness landscape model that harbors epistasis and is consistent with commonly observed patterns of adaptive evolution

In relating genotypes to fitness, models of adaptation need to both be computationally tractable and qualitatively match observed data. One reason that tractability is not a trivial problem comes from a combinatoric problem whereby no matter in what order a set of mutations occurs, it must yield the same fitness. We refer to this as the bookkeeping problem. Because of their commutative property, the simple additive and multiplicative models naturally solve the bookkeeping problem. However, the fitness trajectories and epistatic patterns they predict are inconsistent with the patterns commonly observed in experimental evolution. This motivates us to propose a new and equally simple model that we call stickbreaking. Under the stickbreaking model, the intrinsic fitness effects of mutations scale by the distance of the current background to a hypothesized boundary. We use simulations and theoretical analyses to explore the basic properties of the stickbreaking model such as fitness trajectories, the distribution of fitness achieved, and epistasis. Stickbreaking is compared to the additive and multiplicative models. We conclude that the stickbreaking model is qualitatively consistent with several commonly observed patterns of adaptive evolution.     

Using an optimization algorithm to establish a network of video surveillance for the protection of Golden Camellia

The natural environment is facing increasing human disturbance. Many species of flora are extinct or endangered. To improve the efficiency of ecological management and monitoring, this study proposed to establish a video monitoring network to protect a world-famous rare flora: Golden Camellia, in Fangcheng nature reserve, Guangxi Province, China. Based on the model of LSCP (location set covering problem), we attempted to establish full monitoring coverage of camellias while minimizing the number of video cameras. The model was solved by integer programming. In case of multiple solutions, this study proposed two additional criteria, maximize monitoring area and maximize overlapping count, to eliminate suboptimal solutions. Both of the two optimal solutions included 80 cameras covering a monitoring area of over 55 km². Together, these cameras are able to monitor 97.2% of golden camellia in the reserve. The findings of this study suggest that this location optimization model can be used to improve the conservation effectiveness of rare species.     

Mathematical modeling of induced foreign protein production by recombinant bacteria

A new generalized mathematical model for recombinant bacteria which includes inducer effects on cell growth and foreign protein production is developed. The model equation set was applied to a host-vector system, Escherichi coli D1210 and plasmid pSD8. Batch experiments were designed and performed in shake flasks to verify the model. A parameter estimation method was developed and proven to be efficient. Although simple, the model can effectively describe the dynamics of the production of foreign protein in recombinant bacteria and can be used for optimization and control studies to maximize foreign protein production.     

Extending the range of additivity in using inclusive fitness

Inclusive fitness is a concept widely utilized by social biologists as the quantity organisms appear designed to maximize. However, inclusive fitness theory has long been criticized on the (uncontested) grounds that other quantities, such as offspring number, predict gene frequency changes accurately in a wider range of mathematical models. Here, we articulate a set of modeling assumptions that extend the range of scenarios in which inclusive fitness can be applied. We reanalyze recent formal analyses that searched for, but did not find, inclusive fitness maximization. We show (a) that previous models have not used Hamilton''s definition of inclusive fitness, (b) a reinterpretation of Hamilton''s definition that makes it usable in this context, and (c) that under the assumption of probabilistic mixing of phenotypes, inclusive fitness is indeed maximized in these models. We also show how to understand mathematically, and at an individual level, the definition of inclusive fitness, in an explicit population genetic model in which exact additivity is not assumed. We hope that in articulating these modeling assumptions and providing formal support for inclusive fitness maximization, we help bridge the gap between empiricists and theoreticians, which in some ways has been widening, demonstrating to mathematicians why biologists are content to use inclusive fitness, and offering one way to utilize inclusive fitness in general models of social behavior.