Optimality In Biology

Optimality In Biology – a comprehensive 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.

Examples

(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 – Optimality and evolutionary tuning of the expression level of a protein. [2, 3, 4]: 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.

• Growth rate on different carbon sources - Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth [5]: 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.

• The genetic code - Evolution and multilevel optimization of the genetic code [6]: The discovery of the genetic code was one of the most important advances of modern biology. But there is 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. In this issue, Itzkovitz and Alon show 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.

• 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. [7]: 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.

• Chemotactic response function - The bacterial chemotactic response reflects a compromise between transient and steady-state behavior. [8, 9]: 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.

• Neuronal wiring - Wiring optimization can relate neuronal structure and function. [10]: 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.

• Foraging strategy - Optimal Foraging: A Selective Review of Theory and Tests. [11]: 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.

• 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.

• Life history properties: Age of reproductive maturity, number of eggs in a clutch, etc. [14]
• 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
• morphogen gradients
• 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 lost
• 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
• 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

References:

1. Parker G.A. and Smith J.M. Optimality theory in evolutionary biology. Nature 1990 27-33.

2. Dekel E and Alon U. Optimality and evolutionary tuning of the expression level of a protein. Nature. 2005 Jul 28;436(7050):588-92. DOI:10.1038/nature03842 |
3. ISBN:9781584886426

4. Tomer Kalisky, Erez Dekel and Uri Alon Cost–benefit theory and optimal design of

   gene regulation functions Phys. Biol. 4 (2007) 229–245

5. Ibarra RU, Edwards JS, and Palsson BO. Escherichia coli K-12 undergoes adaptive evolution to achieve in silico predicted optimal growth. Nature. 2002 Nov 14;420(6912):186-9. DOI:10.1038/nature01149 |
6. Bollenbach T, Vetsigian K, and Kishony R. Evolution and multilevel optimization of the genetic code. Genome Res. 2007 Apr;17(4):401-4. DOI:10.1101/gr.6144007 |
7. Vasil'ev S and Bruce D. Optimization and evolution of light harvesting in photosynthesis: the role of antenna chlorophyll conserved between photosystem II and photosystem I. Plant Cell. 2004 Nov;16(11):3059-68. DOI:10.1105/tpc.104.024174 |
8. Clark DA and Grant LC. The bacterial chemotactic response reflects a compromise between transient and steady-state behavior. Proc Natl Acad Sci U S A. 2005 Jun 28;102(26):9150-5. DOI:10.1073/pnas.0407659102 |

9. SP Strong, B Freedman, W Bialek, R Koberle Adaptation and optimal chemotactic strategy for E. coli. - Physical Review E, 1998

10. Chen BL, Hall DH, and Chklovskii DB. Wiring optimization can relate neuronal structure and function. Proc Natl Acad Sci U S A. 2006 Mar 21;103(12):4723-8. DOI:10.1073/pnas.0506806103 |

11. G. H. Pyke, H. R. Pulliam, E. L. Charnov Optimal Foraging: A Selective Review of Theory and Tests. The Quarterly Review of Biology, Vol. 52, No. 2 (1977), pp. 137-154

12. Komarova NL and Wodarz D. The optimal rate of chromosome loss for the inactivation of tumor suppressor genes in cancer. Proc Natl Acad Sci U S A. 2004 May 4;101(18):7017-21. DOI:10.1073/pnas.0401943101 |
13. Komarova N. Does cancer solve an optimization problem?. Cell Cycle. 2004 Jul;3(7):840-4.
14. ISBN:0198577419