# Optimality In Biology – a collection of annotated examples

## Motivation and definition

Optimality – the property of a system to maximize or minimize some function under given constraints – has been a central concept in many fields such as physics, computer science and engineering. In the realm of biology, natural selection leads to exquisite functional life forms all abiding to the laws of physics and chemistry yet show remarkable adaptation to the surrounding conditions. One manifestation of this process is that some characteristics of organisms can be shown to be close to optimally adapted to the constraints of their environment. This website and annotated collection aims to serve as a source of examples that will help discuss and disseminate this form of studying biological processes and inspire the analysis of other biological phenomena using these tools and perspectives.

In many respects the emphasis is on the constrains rather than on the issue of optimality per se, as eloquently framed by Parker and Maynard-Smith: “Optimization models help us to test our insight into the biological constraints that influence the outcome of evolution. They serve to improve our understanding about adaptations, rather than to demonstrate that natural selection produces optimal solutions.” [1]. We encourage everyone interested in this fascinating subject to add examples, comments and join the discussion either by directly editing these pages or by communicating them through email (ron_milo@hms.harvard.edu) and we will add them.

## Optimality and random drift

It is well appreciated that neutral drift has a central importance in the dynamics of properties such as gene frequencies in a population as discussed in the inspiring work of Kimura. There are many other cases where neutral drift in properties and the importance of randomness are established. How does this fit with the concept of optimality in biological systems? The effort to analyze biological systems based on principles of optimality does not mean that they are all optimal or that neutral drift is not the dominant force in many cases. It is a quantitative matter whether in any specific system, affected by both selection and drift, which factor will be dominant. It is suggested that by performing optimality analysis one could find in which cases selection is the dominant driving force and by inference where drift may be the driving force.

## Non optimality in biology

We also aim to collect examples of cases where biological systems are non optimal. This is not a simple task, as we never know if we took all constraints or goal functions into account and thus there is no way to “prove” non-optimality. Yet, there are cases where there is strong evidence that things might not be optimal and result from neutral drift or “frozen” historical accident. For example Way & Silver [2] give the following cases: "The panda’s thumb (Gould, 1980), the placement of the windpipe in front of the esophagus (so that food can go down the wrong tube), traversal of the urethra through the prostate gland (so that if the prostate becomes inflamed and swells, it becomes difficult to urinate)". Similar is the case of the path of the Vas deferens in male anatomy. An interesting case is the anatomy of the human eye which shows exquisite design features but where the rods and cones are located behind the neurons rather than in front as in the octopus eye which leads to the necessity of a blind spot which is suggested to be a case of non optimality.

We will try to collect such examples and discuss them here as well. Please let us know if you have a convincing example.

# Examples - molecular

(we aim to have a concise description of what was achieved in each set of examples. Currently there is the abstract of the respective papers):

## Level of protein expression

Level of protein expression – Optimality and evolutionary tuning of the expression level of a protein. [3, 4, 5]: Different proteins have different expression levels. It is unclear to what extent these expression levels are optimized to their environment. Evolutionary theories suggest that protein expression levels maximize fitness, but the fitness as a function of protein level has seldom been directly measured. To address this, we studied the lac system of Escherichia coli, which allows the cell to use the sugar lactose for growth. We experimentally measured the growth burden due to production and maintenance of the Lac proteins (cost), as well as the growth advantage (benefit) conferred by the Lac proteins when lactose is present. The fitness function, given by the difference between the benefit and the cost, predicts that for each lactose environment there exists an optimal Lac expression level that maximizes growth rate. We then performed serial dilution evolution experiments at different lactose concentrations. In a few hundred generations, cells evolved to reach the predicted optimal expression levels. Thus, protein expression from the lac operon seems to be a solution of a cost-benefit optimization problem, and can be rapidly tuned by evolution to function optimally in new environments.

## Enzyme catalysis

Optimization of the catalytic rate and the ratio Kcat/Km [6, 7]: There is a diffusion limit to the value of Kcat/Km and these references discuss the attainment of this limit in some well researched enzymes such as triosephosphate isomerase (TIM). Fersht's book also makes claims about the relationship between Km and the substrate concentration ([S]) for achievement of the maximal catalytic rate Km>>[S]

(see Fersht book pp. 349-355, 362-368 and as background pp. 54-58, 103-110, 158-168).

## Metabolism

Optimization of metabolism : the evolution of metabolic pathways toward simplicity through the game of the pentose phosphate cycle [8]: Previous theoretical studies on the pentose phosphate cycle (Melendez-Hevia et al., 1985, 1988, 1990) demonstrated that simplicity in metabolism, defined as the least possible number of enzyme reactions in a pathway, has been a target in biological evolution. Those results demonstrated that a process of optimization has occurred in the evolution of metabolism. However, the results also suggest a number of questions of general interest: (i) Why simplicity? What is the selective advantage of simplicity in metabolic pathways? (ii) How has simplicity been achieved? Can natural selection mechanisms solve the problems of combinatorial optimization in the design of metabolism? (iii) Are the reaction mechanisms of the pentose phosphate cycle (transketolase and transaldolase) the best suited for pentose-hexose interconversion

## Growth rate on different carbon sources

Growth rate on different carbon sources - Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth [9]: Annotated genome sequences can be used to reconstruct whole-cell metabolic networks. These metabolic networks can be modeled and analyzed (computed) to study complex biological functions. In particular, constraints-based in silico models have been used to calculate optimal growth rates on common carbon substrates, and the results were found to be consistent with experimental data under many but not all conditions. Optimal biological functions are acquired through an evolutionary process. Thus, incorrect predictions of in silico models based on optimal performance criteria may be due to incomplete adaptive evolution under the conditions examined. Escherichia coli K-12 MG1655 grows sub-optimally on glycerol as the sole carbon source. Here we show that when placed under growth selection pressure, the growth rate of E. coli on glycerol reproducibly evolved over 40 days, or about 700 generations, from a sub-optimal value to the optimal growth rate predicted from a whole-cell in silico model. These results open the possibility of using adaptive evolution of entire metabolic networks to realize metabolic states that have been determined a priori based on in silico analysis.

## Why only four Letters in the genetic alphabet

Four Letters in the Genetic Alphabet: A Frozen Evolutionary Optimum? [10]: The author aims to understand why there are four letters in the genetic code. It analyzes a framework where the fitness in the primordial RNA world depended on the product of the fidelity and the overall reproduction rate. The fidelity is suggested to decrease exponentially with the number of letters and the reproduction rate that depends on the catalytic rate of ribozymes increase sub exponentially. There is thus a maximum that for a large parameter range suggests an optimal fitness at four letters. This optimum was relevant to the RNA dominated world and currently remains as a frozen evolutionary optimum.

Abstract: Piccirilli et al. (Nature, Lond. 343, 33-37 (1990)) have shown experimentally that the replicatable introduction of new base pairs into the genetic alphabet is chemically feasible. The fact that our current genetic alphabet uses only two base pairs can be explained provided that this basic feature of organisms became fixed in an RNA world utilizing ribozymes rather than protein enzymes. The fitness of such riboorganisms is determined by two factors: replication fidelity and overall catalytic efficiency (basic metabolic or growth rate). Replication fidelity is shown to decrease roughly exponentially, and catalytic efficiency is shown to increase with diminishing returns, with the number of letters for a fixed genome length; hence their product, i.e. fitness, gives rise to a set of values with an optimum. Under a wide range of parameter values the optimum rests at two base pairs. The chemical identity of the particular choice in our genetic alphabet can also be rationalized. This optimum is considered frozen, as currently the dominant catalysts are proteins rather than RNAs.

## The genetic code

The genetic code is a dictionary that maps the sixty-four codons, written in the language of nucleic bases, to the language of the twenty amino-acids (1). Each of the 64 molecular symbols in this dictionary carries a molecular “meaning” which is an amino-acid or a stop signal. Thus the design of an optimal code is a semantic challenge of wisely assigning meanings to symbols (2). Observing the code-table, order is evident (3). Firstly, the code is degenerate: only 20 amino-acids are encoded, much less than the maximal possible number of 64. Secondly, codons that are adjacent in the table (.i.e. differing in one base) tend to encode either the same amino acid or chemically similar amino acids (4). Therefore, if one draws a topographical map in which the horizontal coordinate is the location of the codon in the code-table and the altitude is a chemical property of the amino-acid, such as polarity, then the resulting landscape will be smooth. Are the apparent smoothness and degeneracy of the code-table with its twenty amino-acids hallmarks of biological optimality? - Recent research suggests that the answer is affirmative: The genetic code is depicted as a noisy information channel. Errors in the channel, such as misreading a codon or mischarging a tRNA, may distort the mapping from codons to amino-acids. The fitter codes are the ones which minimize this distortion (2). Organisms compete by the fitness of their codes and, as a result, a genetic code emerges at a transition in the noisy channel, when the mapping of codons to amino-acids becomes nonrandom. Moreover, the emergent code is smooth, since smooth codes minimize the distortion at a minimal cost of resources. Thus smooth codes optimize the fitness of the organism. The emergence of the code is governed by the topology of the error-graph, in which codons are connected if they are likely to be confused (5). This topology sets an upper bound on the number of possible amino-acids. The suggested scenario is generic and may describe a mechanism for the formation of other error-prone molecular codes (6, 7), such as the recognition of DNA sites by proteins in the transcription regulatory network (8, 9). However, resilience to errors is only fundamental optimality level in the genetic code. There are many possible genetic codes that provide more or less the same level of resilience, yet only one configuration was chosen to be the universal genetic code. Recent research suggests that this choice is not accidental (10): There is much more to a DNA code than protein sequence; DNA carries signals for splicing, localization, folding, and regulation that are often embedded within the protein-coding sequence. It was shown that the specific 64-to-20 mapping found in the genetic code may have been optimized for permitting protein-coding regions to carry this extra information and suggest that this property may have evolved as a side benefit of selection to minimize the negative effects of frameshift errors.

(contributed by T. Tlusty, Weizmann institute)

1. Crick, F.H. The origin of the genetic code. J. Mol. Biol. 38, 367-79 (1968).

2. Tlusty, T. A model for the emergence of the genetic code as a transition in a noisy information channel. J. Theor. Biol. 249, 331-342. (2007).

3. Woese, C.R. Order in the genetic code. Proc. Natl. Acad. Sci. U. S. A. 54, 71-5 (1965).

4. Sella, G. & Ardell, D. The Coevolution of Genes and Genetic Codes: Crick’s Frozen Accident Revisited. J. Mol. Evol. 63, 297-313 (2006).

5. Eckmann, J.-P. Trading codes for errors. Proc. Natl. Acad. Sci. U. S. A. 105, 8165-8166 (2008).

6. Tlusty, T. Rate-Distortion Scenario for the Emergence and Evolution of Noisy Molecular Codes. Phys. Rev. Lett. 100, 048101-4 (2008).

7. Tlusty, T. A simple model for the evolution of molecular codes driven by the interplay of accuracy, diversity and cost. Physical Biology 5, 016001 (2008).

8. Itzkovitz, S., Tlusty, T. & Alon, U. Coding limits on the number of transcription factors. BMC Genomics 7, 239 (2006).

9. Shinar, G., Dekel, E., Tlusty, T. & Alon, U. Rules for biological regulation based on error minimization. Proc. Natl. Acad. Sci. U. S. A. 103, 3999-4004 (2006).

10. Itzkovitz, S. & Alon, U. The genetic code is nearly optimal for allowing additional information within protein-coding sequences. Genome Res. 17, 405-412 (2007).

## Photosynthetic antenna

Photosynthetic antenna - Position and orientation of the chlorophyll pigments - Optimization and evolution of light harvesting in photosynthesis: the role of antenna chlorophyll conserved between photosystem II and photosystem I. [11]: The efficiency of oxygenic photosynthesis depends on the presence of core antenna chlorophyll closely associated with the photochemical reaction centers of both photosystem II (PSII) and photosystem I (PSI). Although the number and overall arrangement of these chlorophylls in PSII and PSI differ, structural comparison reveals a cluster of 26 conserved chlorophylls in nearly identical positions and orientations. To explore the role of these conserved chlorophylls within PSII and PSI we studied the influence of their orientation on the efficiency of photochemistry in computer simulations. We found that the native orientations of the conserved chlorophylls were not optimal for light harvesting in either photosystem. However, PSII and PSI each contain two highly orientationally optimized antenna chlorophylls, located close to their respective reaction centers, in positions unique to each photosystem. In both photosystems the orientation of these optimized bridging chlorophylls had a much larger impact on photochemical efficiency than the orientation of any of the conserved chlorophylls. The differential optimization of antenna chlorophyll is discussed in the context of competing selection pressures for the evolution of light harvesting in photosynthesis.

- Sener et al. Comparison of the light-harvesting networks of plant and cyanobacterial photosystem I. Biophys J (2005) vol. 89 pp. 1630-1642

- Vasil'ev et al. Optimization and evolution of light harvesting in photosynthesis: The role of antenna chlorophyll conserved between photosystem II and photosystem I. Plant Cell (2004) vol. 16 pp. 3059-3068

- Yang et al. Energy transfer in photosystem I of cyanobacteria Synechococcus elongatus: Model study with structure-based semi- empirical Hamiltonian and experimental spectral density. Biophys J (2003) vol. 85 (1) pp. 140-158

- Sener et al. Robustness and optimality of light harvesting in cyanobacterial photosystem I. J Phys Chem B (2002) vol. 106 pp. 7948-7960)

## Chromosome loss rate

The optimal rate of chromosome loss in cancer [12, 13]: Many cancers are characterized by a high degree of aneuploidy, which is believed to be a result of chromosomal instability (CIN). The precise role of CIN in cancer is still the matter of a heated debate. We present a quantitative framework for examining the selection pressures acting on populations of cells and weigh the "pluses" and "minuses" of CIN from the point of view of a selfish cell. We calculate the optimal rate of chromosome loss assuming that cancer is initiated by inactivation of a tumor suppressor gene followed by a clonal expansion. The resulting rate, p* = 10(-2)per cell division per chromosome, is similar to that obtained experimentally by Lengauer et al. (1997). Our analysis further suggests that CIN does not arise simply because it allows a faster accumulation of carcinogenic mutations. Instead, CIN must arise because of alternative reasons, such as environmental factors, epigenetic events, or as a direct consequence of a tumor suppressor gene inactivation. The increased variability alone is not a sufficient explanation for the presence of CIN in the majority of cancers.

# Examples - behavioral

(we aim to have a concise description of what was achieved in each set of examples. Currently there is the abstract of the respective papers):

## Chemotactic behavior

Chemotactic response function - The bacterial chemotactic response reflects a compromise between transient and steady-state behavior. [14, 15]: Swimming bacteria detect chemical gradients by performing temporal comparisons of recent measurements of chemical concentration. These comparisons are described quantitatively by the chemotactic response function, which we expect to optimize chemotactic behavioral performance. We identify two independent chemotactic performance criteria: In the short run, a favorable response function should move bacteria up chemoattractant gradients; in the long run, bacteria should aggregate at peaks of chemoattractant concentration. Surprisingly, these two criteria conflict, so that when one performance criterion is most favorable, the other is unfavorable. Because both types of behavior are biologically relevant, we include both behaviors in a composite optimization that yields a response function that closely resembles experimental measurements. Our work suggests that the bacterial chemotactic response function can be derived from simple behavioral considerations and sheds light on how the response function contributes to chemotactic performance.

## Foraging strategy

Foraging strategy - Optimal Foraging: A Selective Review of Theory and Tests. [16]: Beginning with Emlen (1966) and MacArthur and Pianka (1966) and extending through the last ten years, several authors have sought to predict the foraging behavior of animals by means of mathematical models. These models are very similar,in that they all assume that the fitness of a foraging animal is a function of the efficiency of foraging measured in terms of some "currency" (Schoener, 1971) -usually energy- and that natural selection has resulted in animals that forage so as to maximize this fitness. As a result of these similarities, the models have become known as "optimal foraging models"; and the theory that embodies them, "optimal foraging theory." The situations to which optimal foraging theory has been applied, with the exception of a few recent studies, can be divided into the following four categories: (1) choice by an animal of which food types to eat (i.e., optimal diet); (2) choice of which patch type to feed in (i.e., optimal patch choice); (3) optimal allocation of time to different patches; and (4) optimal patterns and speed of movements. In this review we discuss each of these categories separately, dealing with both the theoretical developments and the data that permit tests of the predictions. The review is selective in the sense that we emphasize studies that either develop testable predictions or that attempt to test predictions in a precise quantitative manner. We also discuss what we see to be some of the future developments in the area of optimal foraging theory and how this theory can be related to other areas of biology. Our general conclusion is that the simple models so far formulated are supported are supported reasonably well by available data and that we are optimistic about the value both now and in the future of optimal foraging theory. We argue, however, that these simple models will requre much modification, espicially to deal with situations that either cannot easily be put into one or another of the above four categories or entail currencies more complicated that just energy.

## Sensorimotor control

Optimality principles in sensorimotor control [17]: The sensorimotor system is a product of evolution, development, learning, adaptation – processes that work on different time scales to improve behavioral performance. Consequenly, many theories of motor function are based on the notion of optimal performance: they quantify the task goals, and apply the sophisticated tools of optimal control theory to obtain detailed behavioral predictions. The resulting models, although not without limitations, has explained a wider range of empirical phenomena than any other class of models. Traditional emphasis has been on optimizing average trajectories while ignoring sensory feedback. Recent work has redefined optimality on the level of feedback control laws, and focused on the mechanisms that generate behavior online. This has made it possible to fit a number of previously unrelated concepts and observations into what may become a unified theoretical framework for interpreting motor function. At the heart of the framework is the relationship between high-level goals, and the realtime sensorimotor control strategies most suitable for accomplishing those goals.

## Bird migration

Pareto front analysis of flight time and energy use in long-distance bird migration [18]: Optimality models are frequently used in studies of long distance bird migration to help understand and predict migration routes, stopover strategies and fuelling behaviour in a spatially varying environment. These models typically evaluate bird behaviour by focusing on a single optimization currency, such as total migration time or energy-use, without explicitly considering trade-offs between the involved objectives. In this paper, we demonstrate that this classic single-objective approach downplays the importance of variability in bird behaviour. In the light of these considerations, we therefore propose to use a full multi-criteria optimization method to isolate the set of non-dominated, efficient or Pareto optimal solutions. Unlike single-objective optimization where there is only one combination of bird behaviour maximizing fitness, the Pareto solution set represents a range of optimal solutions to conflicting objectives. Our results demonstrate that this multi-objective approach provides important new ways of analyzing how environmental factors and behavioural constraints have driven the evolution of migratory behaviour.

# Examples - physiological and anatomical

(we aim to have a concise description of what was achieved in each set of examples. Currently there is the abstract of the respective papers):

## Neuronal wiring

Neuronal wiring - Wiring optimization can relate neuronal structure and function. [19]: We pursue the hypothesis that neuronal placement in animals minimizes wiring costs for given functional constraints, as specified by synaptic connectivity. Using a newly compiled version of the Caenorhabditis elegans wiring diagram, we solve for the optimal layout of 279 nonpharyngeal neurons. In the optimal layout, most neurons are located close to their actual positions, suggesting that wiring minimization is an important factor. Yet some neurons exhibit strong deviations from "optimal" position. We propose that biological factors relating to axonal guidance and command neuron functions contribute to these deviations. We capture these factors by proposing a modified wiring cost function.

## Leaf size

Optimal Leaf Size in Relation to Environment [20]: The principle of optimal design (Rosen 1967) can be stated as follows. Natural selection leads to organisms having a combination of form and function optimal for growth and reproduction in the environments in which they live.' This principle provides a general framework for the study of adaptation in plants and animals. The efficiency of water use by plants (Slatyer 1964) can be defined as grams of carbon dioxide assimilated per gram of water lost. Leaf temperatures, transpiration rates, and water-use efficiencies can be calculated for single leaves using well-established principles of heat and mass transfer. The calculations are complex, however, depending on seven independent variables such as air temperature, humidity and stomatal resistance. The calculations can be treated as artificial data in factorial design experiment. This technique is used to compare the sensitivity of the system response variables to changes in the independent variables, and to their interactions. The assumption is made (as a first approximation) that the optimal leaf size in a given environment is the size yielding the maximum water-use efficiency. This very simple assumption leads to predictions of trends in leaf size which agree well with the observed trends in diverse regions (tropical rainforest, desert, arctic, etc.). Specifically, the model predicts that large leaves should be selected for only in warm or hot environments with low radiation (e.g. forest floors in temperate and tropical regions). There are some plant forms and microhabitats for which observed leaf sizes disagree with the predictions of the simple model. Refinements are thus proposed to include more factors in the model, such as the temperature dependence of net photosynthesis. It is shown that these refinements explain much of the lack of agreement of the simpler model. One of the main roles of mathematical models in science is to pose sharp questions' (Kac 1969). The present model suggests several speculative propositions, some of which would be difficult to prove experimentally. Others, whether true or not, can serve as a theoretical framework against which to compare experimental results. The propositions are as follows. (1) Every environment tends to select for leaf sizes increasing the efficiency of water utilization, that is, the ratio of CO_2 uptake to water loss. (2) Herbs are physiologically different from woody plants, in such a way that water-use efficiency has been more important in the evolution of the latter. (3) The stomatal resistance of a given leaf varies diurnally in such a way that the water-use efficiency of that leaf tends to be a maximum. (4) The larger the photosynthesizing surface of a desert succulent, the more likely it is to exhibit acid metabolism, with stomata open at night and closed during the day. (5) In arctic and alpine regions, the plant species whose carbohydrate metabolism is most severely limited by low temperatures are most likely to evolve a cushion form of growth. In addition to providing these testable hypotheses, the results of the model may be useful in other ways. For example, they should help plant breeders to alter water-use efficiencies, and they could help palaeobotanists interpret past climates from fossil floras.

Also, see discussion in David Gates, Biophysical ecology, pp. 369-374

# More examples

• Life history properties: Age of reproductive maturity, number of eggs in a clutch, etc. [21]
• Vascular branching - Murray's law stating that in a branching from a vessel of radius r1 to two vessels of radius r2 and r3 there is a relationship between them where r1^3=r2^3+r3^3. This can derived based on the assumption that the vascular system is built in a way that minimizes the combined cost of construction and maintainance. The maintainance is governed by the drag that according to Poiseuille's law goes as r^-4. (Rosen 1967 book Ch. 3, more Ref.)
• Glycogen as energy storage vehicle (Cornish Bowden 2004 book Ch. 6, E Melendez-Hevia refs.)
• Metabolism - the pentose phosphate pathway and the Calvin cycle (Cornish Bowden 2004 book Ch. 5, E Melendez-Hevia refs.) - The steps in these two pathways perform transformations with some well defined constraints. The set of reactions seen in Nature is the shortest path from the substrates to the products given these constraints. A very accessible discussion is given in Cornish-Bowden's book.
• Metabolism - Glycolysis Theoretical approaches to the evolutionary optimization of glycolysis--chemical analysis. PMID: 9119021 Theoretical approaches to the evolutionary optimization of glycolysis: thermodynamic and kinetic constraints. PMID: 9030739
• Codon usage and biases (Ref.)
• Shapes that minimize drag (Ref. fish, fungi spores, birds?)
• Prey interception strategy of bats (Ghose et al., PLOS Biology 2006)
• Optimal virulence level (Jensen et al., PLOS Biology 2006)
• Neural information transmission (Bialek 1997)
• tRNA levels - "Growth rate-optimised tRNA abundance and codon usage" Otto G. Berga, C. G. Kurlanda Journal of Molecular Biology Volume 270, Issue 4, 25 July 1997, Pages 544-550,
• photosynthesis wavelength
• enzymes near the diffusion limit
• Optimal metabolic network operation (“Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli” Robert Schuetz, Lars Kuepfer1 & Uwe Sauer Also Jens Nielsen, 2007)
• Neuronal energetics - Simon Laughlin
• Compound eye aperture size at the diffraction/resolution limit
• Water usage efficiency - the opening of the stomata for maximal carbon fixation per water loss
• Allocating leaf nitrogen - Allocating leaf nitrogen for the maximization of carbon gain: Leaf age as a control on the … C Field - Oecologia, 1983 - Springer; Maximizing daily canopy photosynthesis with respect to the leaf nitrogen allocation pattern in the,T Hirose, MJA Werger - Oecologia, 1987 - Springer; Trade-off Between Light-and Nitrogen-use Efficiency in Canopy Photosynthesis T HIROSE, FA BAZZAZ - Annals of Botany, 1998 - Annals Botany Co; Photosynthesis and nitrogen relationships in leaves of C 3 plants - JR Evans - Oecologia, 1989 - Springer; RESOURCE LIMITATION IN PLANTS AN ECONOMIC ANALOGY - AJ Bloom, FS Chapin III, HA Mooney - Ann. Rv. Ecol. Syst, 1985 - Annual Reviews
• branching structure of the vascular tree - Optimal branching structure of the vascular tree, A Kamiya, T Togawa, Bulletin of Mathematical Biology, 1972
• Optimal gene partition into operons correlates with gene functional order - Phys. Biol. 3 (2006) 183–189
• Demand theory/error load minimization - Savageau; Shinar
• metabolism and fluxes - Principles of optimal metabolic network operation, Jens Nielsen; Schuetz R, Kuepfer L, Sauer U (2007) Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli. Mol Syst Biol 3: 119
• other - A theory of optimal differential gene expression W Liebermeister, E Klipp, S Schuster, R Heinrich - BioSystems, 2004 - Elsevier
• phage life history - http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WMD-4JRKJKC-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f38b4366cb6799e758eaf5bac1f6252f
• litter size in birds - Lack (1947), Owen (1983), Perrins & Moss (1975), Perrins & Birkhead (1983).
• plant resource allocation for leafs and petioles - Pearcy and Yang (1998) that demonstrated an optimal partitioning between leaves and petioles, maximizing light capture and C gain in the redwood forest understory rosette plant, Adenocaulon bicolor. It also is similar to the approach of Takenaka et al . (2001), who found that allocation to petioles in an understory palm Licuala arbuscula was optimal for light capture.

# Books

## Optimality Principles in Biology - Robert Rosen

Optimality Principles in Biology, Robert Rosen, Butterworths, London, 1967

## The Pursuit of Perfection - Athel Cornish-Bowden

The Pursuit of Perfection - Aspects of Biochemical Evolution, Athel Cornish-Bowden, Oxford University Press, 2004

In The Pursuit of Perfection the author explains how the biochemical processes that occur in living cells, long thought to be evidence of intelligent design rather than evolution, are now understood to be the result of natural selection. This book will principally appeal to senior undergraduates, graduates and scientists but the style, content and organisation of the book are intended to make the book accessible for all scientifically-minded individuals.

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[Parker-Nature-1990]
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gene regulation functions Phys. Biol. 4 (2007) 229–245

[Kalisky-PhysicalBiology-2007]
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[Knowles-ACR-1977]
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[Fersht-Book-1998]
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[Melendez-JTB-1994]
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[Strong-PRE-1998]
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