Research Interests

This page provides a third-person narrative of my research interests and accomplishments. A PDF version includes a brief introductory biography.

Overview

Frank uses mathematical and computational models to study the natural processes that shape the design of organisms. His research spans genome evolution, social evolution, infectious disease, cancer evolution, and the mathematical foundations of natural selection theory. His work also connects evolutionary dynamics with principles from information theory, machine learning, and physics.

Research topics

Frank’s research began with empirical fieldwork on fig wasps and has since expanded into theoretical work across several areas of evolutionary biology. Recurring themes include the natural processes that shape the design of organisms and how design fails in disease. His 2007 book on cancer¹ emphasizes that failure reveals design, linking his studies of disease to his broader interests in the adaptations of organisms.

His 1996 article The design of natural and artificial adaptive systems proposes a broad synthesis between adaptive biological and engineering systems². He connects immunology, the design of intelligent computer systems and robots, the ways in which information is stored, mixed, and transmitted, and the processes that wire brains in development, among other topics. The common theme is that learning and natural selection form a unified foundation for the study of many great problems in the history of life and in human advances in knowledge and engineering.

Frank’s work often moves between specific biological puzzles and general theory. For example, the contrasting age onset patterns of cancer in different tissues exposed gaps in existing theory and motivated new conceptual approaches¹. Those approaches in turn generated testable predictions, such as how specific aspects of genetic control over development alter the origins of cancer³.

The theoretical tools developed for individual problems have broader reach, leading to synthetic work that unifies observations and conceptual frameworks across topics^4,5. The move from observed puzzles of biology to broad synthetic theory recurs in his work on genomic conflict, attack and defense, sex ratios, social evolution, learning, robust regulatory control, and commonly observed probability patterns in data.

The following sections describe specific research topics in more detail. The first sections emphasize conflict, cooperation, life history, and social evolution. The later sections turn to attack and defense, immunology, cancer, metabolism, robust regulatory control, general principles of learning and natural selection, and the natural structure in data patterns that shapes observation and inference.

All topics remain within the broader scope of understanding how natural forces shape the design of organisms, with scholarly syntheses that link natural selection to wider understanding in engineering, machine learning, physics, and probability theory.

These links provide quick access to individual topics:

• Early work: fig wasps and sex allocation
• Genome evolution
• Variable environments
• Social evolution, major transitions, and parasite virulence
• Host control of symbionts, germ-soma separation
• Synergistic symbiosis
• Attack and defense, metapopulation dynamics
• Immunology and infectious disease
• Cancer evolution, somatic mosaicism
• Microbial metabolism
• The paradox of robustness
• Robustness and complexity, hourglasses and protocol waists
• Regulatory evolution, engineering principles in biology
• Machine learning, evolutionary processes, and circuit design
• Natural selection, the Price equation, and theoretical unification
• Common patterns, probability, symmetry, and scaling laws

Early work: fig wasps and sex allocation

Frank’s master’s and doctoral research included studies on the behavior, morphology, and ecology of fig wasps, experimental studies of sex ratios in fig wasps, and the theory of sex allocation^6–8. He published a series of papers in the 1980s on hierarchical selection theory and sex ratios, developing models that linked local population structure to the evolution of sex ratio strategies^9,10. The fig wasp species Pegoscapus franki is named after him¹¹.

His graduate work also created new mathematical methods to analyze dispersal, demography, and sex allocation under complex natural histories. Those methods provided a way to find simple solutions in terms of general expressions of kin selection and reproductive value. That work showed how kin selection and group selection reduce to the same underlying logic when properly framed, resolving a long-standing debate^9,12.

His article on Individual and population sex allocation patterns⁸ also cleared up broad confusion about sex allocation. He showed that whenever individuals are favored to modulate their sex allocation among their offspring, Fisher’s¹³ widely cited theory for equal allocation between the sexes does not hold at the population level. Instead, the specific pattern of fitness returns for allocation to the sexes and the distribution of resources among parents link individual and population allocation ratios.

The theory of sex allocation has provided one of the most productive testing grounds for the study of adaptation by natural selection—how natural processes design organisms^12,14–16.

Genome evolution

Frank’s Ph.D. thesis also analyzed genome evolution, addressing how conflicts between different genetic elements within an organism shape reproductive traits. His 1989 paper on cytoplasmic male sterility¹⁷ modeled the evolutionary dynamics of conflict between cytoplasmic and nuclear genes in plants. On one side, mitochondrial genes benefit from sterilizing male function, which redirects resources to seed production and thus to maternal transmission of mitochondria. On the other side, nuclear genes are selected to restore male fertility because they are transmitted equally through sexes. Frank predicted that multiple alternative mitochondrial and nuclear types would create complex, fluctuating spatial dynamics of genomic and ecological conflict, which has matched many empirical studies¹⁸.

His subsequent work on meiotic drive suggested that competition among genetic elements within an organism can initiate speciation by causing hybrid incompatibility^19–21. In particular, meiotic drive between sex chromosomes may explain some aspects of Haldane’s Rule, the observation that among hybrids of recently diverged species the heterogametic sex is often sterile or inviable²². These ideas contributed to the broader genomic conflict theory of speciation, subsequently developed by several groups and now a major area of study^23,24.

With Laurence Hurst, he published a 1996 Nature paper arguing that mitochondrial inheritance creates a systematic bias in disease²⁵. Because mitochondria are transmitted only through mothers, natural selection can remove mitochondrial mutations that harm females but is blind to mutations that harm only males. This transmission asymmetry predicts a class of male-biased diseases caused by mitochondrial dysfunction, possibly explaining some fraction of observed male infertility and other diseases such as Leber’s Hereditary Optic Neuropathy (LHON). Several subsequent studies supported this idea^26,27, and several labs continue to work on this topic^28,29, although some reviews question the broad consequences for particular traits such as aging³⁰.

With Bernie Crespi, Frank published a 2011 PNAS paper on X-autosome conflict arising from the differing selective pressures on males and females (sexual antagonism) and the differing sex-specific transmission patterns of the X chromosome³¹. The theory predicts genome-wide antagonistic regulatory control of traits that have sex-specific consequences for fitness, making such traits easily perturbed. That sensitivity predicts pathologies of extreme expression that push beyond the normal continuum of male–female differences in physiology and behavior, possibly leading to conditions such as autism and psychosis³².

Overall, the work has linked genomic conflict to ecological dynamics, speciation, and disease, showing the broad evolutionary importance of what had initially been considered sporadic oddities in genetics.

Variable environments

With Monty Slatkin, Frank published Evolution in a variable environment³³. Fitness is the driving force of adaptation and evolution, and fitness always varies across time and space. Previously, different aspects of fitness variation had been studied in separate theoretical frameworks, each addressing a particular level or context^34–37.

Frank and Slatkin unified those separate approaches by showing that they arise as different aspects of the underlying hierarchical structure of correlations in fitness. The different roles of ecology, behavior, and the multidimensional architecture of phenotypes can then be understood within a single deeper framework for the causes of variation and their consequences for success. The article also linked this evolutionary framework to the broad theory of economics under risk and uncertainty³⁸, showing that the same mathematical structure governs both biological adaptation and economic decision-making under variable returns.

A later article provided a simple summary of the key ideas, with emphasis on the links to classic economic theory³⁹. A key difference is that biological theory naturally tracks relative fitness rather than absolute returns, analogous to the distinction between market share and absolute success in economics. That perspective highlights aspects of risk and uncertainty that standard economic frameworks tend not to emphasize.

Social evolution, major transitions, and parasite virulence

Tragedy of the commons, repression of competition

In the 1990s, Frank took up a broad puzzle in social evolution. Kin selection seemed not to be a sufficient explanation for suppression of competition among lower-level units in the formation of higher-level units and individuality. For example, what controls the intrinsic tendency for genes to compete within genomes or individuals to compete within extended non-kin groups? This puzzle had particularly been emphasized in the early evolution of cells and in the transition of human sociality from small groups to larger societies^40–42, later synthesized in Maynard Smith and Szathmáry’s book The Major Transitions in Evolution⁴³.

Frank’s 1995 Nature paper, Mutual policing and repression of competition in the evolution of cooperative groups⁴⁴, showed that when self-restraint evolving through kin selection is not sufficient to favor the cooperative integration of groups, natural selection can under certain circumstances favor various mechanisms of mutual policing in which group members enforce reproductive fairness, group integration, and a key step toward the evolution of a higher-level unit of individuality^45,46.

That Nature article presented Frank’s model as a formalization of the tragedy of the commons in an evolutionary context and proposed repression of internal competition as a mechanism to overcome the tragedy. The tragedy of the commons concept was previously mentioned⁴⁷ but rarely cited in the evolutionary literature before the 1995 Nature paper⁴⁸. Soon after, the tragedy model became a common standard in evolutionary analyses of conflict and cooperation^49,50.

Parasite virulence and microbe social evolution

Frank’s 1996 Models of parasite virulence⁵¹ linked the evolutionary puzzle of a parasite’s virulent harm to its host to the broader social evolution concept of the tragedy of the commons. A parasite depends on its host for resources, yet competition between parasites within the host tends to favor overexploitation of the common resource, degrading the success of the group of infecting parasites, a perfect example of the tragedy of the commons.

Starting from those aspects of resource competition and social conflict, Frank developed a broad theoretical framework for how infection dynamics influence the evolution of virulence. The framework integrated kin selection theory, life history theory, and epidemiology, extending prior work into a broader notion of social evolution in microbes. Previous kin selection models rarely accounted for complex demography in the evolution of social traits. But in parasites, the complex demography of infection and transmission, and the tradeoffs between survival and reproduction, are as important as coefficients of relatedness in shaping the evolution of virulence.

The 1996 virulence article emphasized the potential for microbial social evolution as a new and rising field of study, a prediction borne out by the subsequent growth of the field^52,53. Frank’s article also included many applications of the broader theory to specific examples of parasite virulence, a topic extended to more detailed and realistic biological models of infection in a later paper with Paul Schmid-Hempel, Mechanisms of pathogenesis and the evolution of parasite virulence⁵⁴.

New theoretical methods for life history and sociality

Realistic problems such as parasite virulence demanded new theoretical methods, because prior social evolution techniques did not easily account for complex demographies and life histories. In the first step, Frank noted that the standard use of derivatives for the maximization of fitness led to intrinsic expressions that are the coefficients of relatedness in kin selection⁴⁴. That step made the modeling of kin selection simple, automatic, and natural. Express fitness in terms of trait values, take the derivative of fitness with respect to those traits, and recognize the kin selection coefficients that arise from the resulting expressions.

In the second step with Peter Taylor, Frank extended that maximization technique for kin selection by embedding it into a full demographic method for life history and reproductive value. Their 1996 article How to make a kin selection model⁵⁵ provided a complete accounting for the joint effects of demography, life history, and kin selection in the evolution of social traits. Again, the technique followed from the simple use of differentiation in standard maximization methods, making the method relatively easy to use for a wide range of realistic natural histories.

Frank’s two subsequent articles Multivariate analysis of correlated selection and kin selection with an ESS maximization method and The Price equation, Fisher’s fundamental theorem, kin selection, and causal analysis formalized the approach and linked it to other topics^56,57.

Those articles also clarified the distinction between direct and inclusive fitness models of kin selection. Direct fitness tracks the pathways of transmission and heritability of genes that influence social traits. Inclusive fitness reverses the direction of causality in order to assign primary cause for the evolutionary change of traits to the actor that expresses such traits. Although the inclusive fitness notion of causality is attractive, Frank showed that it only works mathematically under particularly simple assumptions about how traits influence outcomes and about the genetic variability of such traits. In essence, inclusive fitness is appealing conceptually but in practice either leads to incorrect results or demands very difficult theoretical adjustments that are nearly always ignored.

His book Foundations of Social Evolution¹² synthesized the new theoretical methods with evolutionary game theory, classical models of natural selection, quantitative genetics, and scholarly links to the methods and history of economics and causal analysis. The book illustrates the value of the methods with applications to problems that combine ecological, demographic, and social processes. Several chapters use the new techniques to unify and extend the theory of sex allocation.

Later publications have returned to these themes^5,58. Recently, Natural selection at multiple scales⁵⁰ unifies many topics, including social evolution, by showing that the evolution of traits is commonly shaped by opposing forces of natural selection acting at different temporal or spatial scales.

Host control of symbionts, germ-soma separation

The advanced understanding of virulence and the tragedy of the commons led to broader insights into symbiosis. For example, hosts sometimes use symbionts to provide nutrients or other services. Frank’s Host-symbiont conflict over the mixing of symbiotic lineages⁵⁹ contrasted host and symbiont interests.

When symbionts compete within the host for their own relative success, symbiont investment in competition against their neighbors tends to degrade their investment in the host’s success. Reduced host success lowers the success for the group of symbionts that live in that host, in the same way that parasite virulence degrades the host resources on which the parasites depend.

Consequently, hosts are favored to reduce competition among their symbionts. One way to do that is to reduce the mixing of different symbiont genotypes within the host. As the relatedness among the symbionts rises, the tendency for competition between symbionts declines, and the interests of host and symbiont converge on the joint success of the cohabitants.

For example, fig trees appear to limit the number of pollinating wasps that enter each fig⁶⁰. Reduced competition between wasps within figs increases the fraction of female offspring produced by the wasps⁷. In this situation, male offspring are a form of competition between different mother wasps because the mother’s sons compete with each other for mates within the fig^7,61–63. From the tree’s point of view, female wasp offspring carry the tree’s pollen to new figs, whereas male wasp offspring provide no benefit. The host trees, by limiting the number of wasps that enter a fig, reduce internal competition between the wasps and raise the benefits of the symbionts to the tree⁵⁹.

Similarly, subsequent studies suggest that ants that farm fungi for food tend to restrict the number of fungal genotypes they use to seed their crops^64,65. Less genetic diversity means less tendency for competition between the fungi for success against neighbors and greater overall productivity, another case of host control over the tragedy of the commons among their symbionts.

In Host control of symbiont transmission: the separation of symbionts into germ and soma⁶⁶, Frank applied the same theory to the widespread vertically transmitted symbionts of insects. Some evidence from classic microscopic studies of these symbionts suggests that the hosts sometimes partition the symbionts within their bodies into subsets. A somatic subset near the gut provides nutrient benefits but is restricted from access to the host’s eggs and subsequent transmission to the next generation. Another subset has access to the germline. That germ-soma separation restricts the opportunity for reproductive competition between the symbionts and, by the repression of competition theory, favors greater unity in the interests and performance of the host-symbiont group⁶⁷.

Synergistic symbiosis

In many cases, symbiotic interactions happen between approximately equal partners rather than between a potentially controlling host and a subordinate symbiont. For example, early molecular replicators near the origin of life may have formed cooperative associations that led to the multiple coordinated functions of early cells⁶⁸. Or, in many bacteria, one sometimes sees positive feedback between the factors excreted by one partner and factors taken up and used by the other, for example, a food source in exchange for an amino acid⁶⁹.

Frank’s article The genetics of mutualism: the evolution of altruism between species⁷⁰, asks: how do such mutualistic interactions arise? The theoretical models in that paper suggest that, initially, there must be genetic correlations between partners from the different species in the tendency to engage in mutualistic exchange. The theory develops analogs between Mendelian recombination and spatial cotransmission of partners and between genetic correlations that are like linkage disequilibrium and also like the coefficients of relatedness from social evolution⁷¹.

The theory led to two conclusions. First, for mutualism to evolve, particular ecological and fitness conditions for the interaction must hold to provide sufficient between-species correlations. Second, the mathematical similarities of linkage disequilibrium, genetic correlations between species, and coefficients of relatedness in kin selection revealed something deeper. If the same processes that we call kin selection can happen between species, then the fundamental understanding of the underlying evolutionary processes must change^56,57,71. Correlations in fitness components are the general process. Kin selection and correlations between partners in different species are instances of that general cause. These insights resurfaced in Frank’s generalization and synthesis in his later book Foundations of Social Evolution¹².

Frank’s The origin of synergistic symbiosis⁷² linked these insights to the processes that cause transitions in the units of selection. In particular, the theory suggested a natural progression from unaligned replicators or species to synergistic partners with positive feedback and genetic correlations between partners to a threshold that, once passed, causes the partners to become obligately dependent on each other. Once obligate dependency arises, there is no longer a need for genetic correlations between partners to favor mutual benefits. Instead, obligate dependency keeps the interests of the partners aligned.

Attack and defense, metapopulation dynamics

Frank broadened his earlier work on cytoplasmic male sterility and genomic conflict¹⁷ to four recurring themes in attack and defense.

Mechanisms of invasion and evasion

In Evolution of host-parasite diversity⁷³, he developed the idea that the specific mechanisms of attack, defense, and recognition would greatly influence the nature of ecological and evolutionary dynamics. For example, there may be genetically based recognition of invasion by defenders and evasion of recognition by attackers. Or physical barriers of defense may be overcome by mechanical penetration. There may be one primary mechanism of defense or many separate ones—the dimensionality of the conflict. If there are multiple mechanisms, they may act in series or in parallel. Each defense mechanism may be quantitatively varying or qualitatively all or none.

Frank showed that the particular mechanistic basis of attack and defense unified understanding of seemingly different cases such as genomic conflict, predator-prey, plant-herbivore, host-pathogen, and so on. In each case, the dimensionality of the traits involved, and whether specificity was qualitative or quantitative, primarily determined diversity and dynamics.

Later articles refined the topics of specificity, recognition, and diversity, and applied those ideas to quantitative traits that are common in plant-herbivore systems, to specific traits that are common in the highly polymorphic bacteriocins of microbial competition, to restriction-modification systems in bacterial defense against viruses, to fungal diseases of plants, and to meiotic drive and cytoplasmic male sterility as examples of genomic conflict^18,74–81.

Genetic and ecological dynamics

Predation, infection, or herbivory often causes large ecological fluctuations in population size of attackers and defenders. Thus, whether different attackers and defenders arise from different genotypes in a particular species or from different species, the overall system follows the dynamics of classical ecological communities.

To build a unified framework for attack and defense, Frank’s articles Ecological and genetic models of host-pathogen coevolution⁸² and Spatial variation in coevolutionary dynamics⁸³ formalized analysis by studying high dimensional ecological community dynamics. The community members may be genotypes or species. Sexual reproduction and Mendelian genetics add a layer of complexity but do not alter the fundamental structure of the system⁷⁵.

Ecological and genetic models of host-pathogen coevolution⁸² used the simplest genetic assumptions for specificity, the “matching alleles” framework, which became a common standard for such models in the subsequent literature⁸⁴. Recognition and polymorphism in host-parasite genetics considered the variety of genetic assumptions and their consequences⁷⁶. The broadly unifying Evolution of host-parasite diversity linked mechanistic assumptions of attack and defense to their genetic and ecological dynamics⁷³.

Potential vs actual diversity & metapopulation dynamics

If there are potentially multiple types of attackers and defenders, then individual spatial locations will often lack particular specificities. For example, a local patch of defenders might not have the specific recognition for certain attack types that are locally absent. Then, a colonization event by that specific attack type can sweep through that local undefended population. That process rapidly alters the local type compositions, driving out other defense specificities in the defenders and attack specificities in the attackers^17,18,85,86.

After such turnover, different attack and defense specificities are favored to colonize that patch. Because such dynamics happen in partially uncorrelated ways over the metapopulation, local diversity tends to be low in each patch because of sweeps by colonists, whereas diversity tends to be high between patches because of the intrinsic tendency for asynchrony in the colonization-extinction processes of genotypes.

That interaction between the dimensionality of the specificity and spatial processes linked the theory to observations on cytoplasmic male sterility^17,18 and on plant diseases¹⁸. That emphasis on nonequilibrium metapopulation dynamics was also promoted by John Thompson and his colleagues^87,88.

One general conclusion was that greater qualitative specificity and more polymorphic alternatives in defense recognition and attack escape from recognition tend to enhance the role of spatial dynamics and global fluctuations in attack success. In ecological terms, as the potential dimensionality of the genotypic community rises in attack and defense, the control of dynamics shifts from local processes to global spatial processes, making a strong prediction about comparative dynamics^17,18,85,86. That prediction links the genetic and physiological mechanisms of particular attack episodes to the global dynamics of populations.

Observation and inference

Many systems of attack and defense have remarkable diversity of specific genotypes^17,75,76. For example, species with cytoplasmic male sterility may have many different mitochondria genotypes, each of which causes male sterility by a different mechanism. We know the mechanisms differ because, for each mitochondrial type, pollen fertility can be restored by different nuclear genes.

Empirical study is challenging in such mutual systems of attack and defense genes. One can only recognize distinct mitochondrial types when they are tested against the presence and absence of the matching nuclear genes. And one can recognize the distinct nuclear genes only when tested against the presence and absence of the matching mitochondrial types.

The literature often contains conflicting reports about the diversity of genotypes in attack-defense systems and in opinions about the dimensionality of such systems. Understanding this issue is important because dimensionality and potential diversity of a system may determine the nature of temporal and spatial dynamics. Frank showed that conflicting reports in the literature often arise from the inherent difficulty of inferring reciprocal diversity, and discussed ways to improve inference in such systems^17,76,89,90.

Immunology and infectious disease

Frank’s publication of A model for the sequential dominance of antigenic variants in African trypanosome infections⁹¹ marked a shift in his interests toward systems in which defense was mediated by more complex immune processes. Individual genotypes of trypanosome parasites store multiple alternative specificities, in which each individual parasite expresses only one of the specificities. Upon initial infection, the host learns to recognize and defend against the initial parasite specificity by adaptive immunity. Meanwhile some of the parasites shift to express a different specificity, temporarily escaping recognition and allowing the infection to continue.

That work triggered broader interest in antigenic variation, the specificities in pathogens and parasites that hosts can learn to recognize and defend against. Attackers alter their antigenic specificities by mutation, recombination, or programmed expression of stored variants, escaping host defense. On the host side, the history of past infections and immune memory set the landscape of defense that temporarily excludes certain attack specificities. However, that immune memory landscape changes as hosts die, newborns enter the population, and infection continues.

Frank’s book Immunology and Evolution of Infectious Disease⁴ considered antigenic variation across pathogens and parasites, including HIV, influenza, foot-and-mouth disease, malaria, and many other systems. The work synthesized the molecular, immunological, and evolutionary processes that together determine the course of disease in populations. Each chapter ends with explicit unsolved puzzles. For example, how does the demography of populations alter the ecological and evolutionary processes that determine antigenic variation in pathogens and immune memory in hosts?

These studies led to collaboration with Alan Barbour on antigenic variation in Borrelia bacteria. Individual cells switch their antigenic type stochastically yet the ordering of antigenic variants expressed by the bacterial population tends to follow a regular pattern. Mechanistic study of the switch mechanism combined with mathematical modeling of immune dynamics within hosts inferred a stochastic hierarchy of antigenic changes within cells that led to the observed pattern of antigenic variant dynamics within hosts⁹².

With Joan Jeffrey, Frank developed the idea of natural vaccination, in which subclinical infections can lead to protection against disease for a fraction of the population⁹³. Their model emphasized cases in which there are two routes of infection, one that is likely to induce immunity but few symptoms, and one that causes significant disease. For example, some pathogens can induce immunity through small cuts in the skin but only cause disease when invading through the nasal passages or lungs. The within-host dynamics of infection by the alternative routes determine the frequency of protection by natural vaccination versus significant disease.

Many zoonotic infections potentially fit this scenario, as outlined in an article with Jeffrey and also in a later 2018 preprint with Robin Bush, Occupational immunity and natural vaccination⁹⁴, suggesting that protection may be common in certain workers in health care or food processing.

In collaboration with Paul Schmid-Hempel, Frank considered the puzzling variability in the number of pathogens required to start a symptomatic infection⁹⁵. They suggested that pathogens such as Shigella, which require a relatively small infective dose, may invade by deploying virulence factors that influence nearby host tissues. With local action, relatively small amounts of secreted virulence factors may be sufficient to establish infection. By contrast, pathogens such as Vibrio cholerae require infection doses many orders of magnitude greater. Such high dosage may be characteristic of pathogens that secrete distantly acting immunomodulatory virulence factors, which would require a large initial invading population to generate sufficient amounts of the essential factors.

In another article, Frank and Schmid-Hempel asked why immune systems often have very many negative regulators of immunity, factors that reduce or turn off an immune response⁹⁶. They proposed that immune systems must respond very quickly to a potential infection, and that need for speed causes a high false positive error rate. To balance those errors, the system requires multiple slower acting and more accurate negative regulators to recognize false alarms and turn off the system. The tension between these strongly opposing positive and negative regulators leads to a delicate balance that is easily perturbed, potentially explaining why immune misregulation may cause substantial disease⁹⁷.

Cancer evolution, somatic mosaicism

Beginning in the early 2000s, Frank published a series of papers analyzing the multistage theory of carcinogenesis and genetic predisposition to cancer^98–104.

His book Dynamics of Cancer: Incidence, Inheritance, and Evolution¹ unified epidemiological patterns of age onset with the mechanistic basis of cancer origin and progression in terms of genetics and tissue physiology. The book comprehensively synthesized incidence data in different tissues stratified by sex, geographic location, and inherited predisposition. Those observations are set within a broad mathematical framework that generated testable predictions for the causes of cancer, organized in a way that provided a new approach to causal understanding.

A series of articles emphasized the role of somatic mutations in early development^105–108. Such developmental mutations create broad mosaicism in mutational distribution throughout the adult body. His work predicted that such mosaicism must be widespread because of the basic mathematics of mutation rates and cell division, and that mosaicism is likely to be a significant cause of adult onset cancer and neurodegeneration. These predictions were made before single-cell genetics was possible. Subsequent empirical work with single-cell technology has broadly supported the likely importance of mosaicism in adult-onset disease for both cancer and neurodegeneration^109,110.

With Marsha Rosner, Frank argued that the extensive stochasticity of cellular behavior and the developmental plasticity of cells likely play strong causal roles in the evolution of resistance to treatment and in the early steps of cancer’s origins¹¹¹. With Itai Yanai, he extended the argument to include the recent advances in single-cell transcriptomics¹¹². The new data showed how key cancer genes, such as KRAS and p53, may primarily influence the origin of cancer by reversing the normal terminal differentiation of cell types in tissues, releasing developmental plasticity to create the novel traits and tissues that make a cancer.

Frank emphasized that this new perspective provides an alternative to the classic oncogene and tumor suppressor gene theory for cancer’s origins. His PNAS article How cancer arises: genetics releases, plasticity creates, and genetics stabilizes³ sets out an alternative three step pathway for the origin of tumors, linking Mary Jane West-Eberhard’s broad theories of developmental plasticity¹¹³ as the driver of novel traits in the history of life with how cancers evolve within bodies. The paper emphasizes that “Cancer is normal development spun out of control.”

Microbial metabolism

Frank’s book Microbial Life History: The Fundamental Forces of Biological Design⁵ links the multiple steps of metabolic biochemical reactions with the processes of natural selection that inevitably shape the fine details of that biochemistry. The synthesis emphasizes economic trade-offs between microbial traits, the thermodynamic structure of metabolism, how natural selection tunes metabolic control through the modulation of thermodynamics, and the demographic structure of populations.

A key puzzle arises from the common observation of overflow metabolism¹¹⁴. Under some conditions, certain microbes take up food sugars quickly, extract a small amount of the free energy, and excrete the biochemical products that still contain most of the usable free energy in the original sugars. Why dump so much of the usable food?

Typically, taking up sugars quickly and dumping usable products associates with faster growth rate but lower yield efficiency, the tradeoff between growing fast (biomass per unit time) and growing efficiently (biomass per unit of food consumed)^115,116. The book synthesizes a variety of theories and empirical studies for how natural selection might tune microbes over that rate vs yield tradeoff. For example, the tradeoff may be set by constraints imposed by limitations on the number of proteins within cells or by limitations imposed by the membrane surface that must take up food and also process that food through the electron transport chain.

The book also emphasizes the variety of design details that natural selection has likely tuned. For example, membrane permeability influences the rate of uptake of food, the ability to maintain the proton motive force used to make ATP, and the defense against attack molecules made by competitors. Different microbes use various sources as their primary food source, ranging from reduced sulfur to methane to the common sugars, each food source inducing unique biochemical tradeoffs. Some species begin digestion of complex carbohydrates extracellularly by secreting digestive enzymes, imposing tradeoffs between external breakdown and loss of digested products to uptake by neighbors.

Other species invest in obtaining the oxidizing molecules that provide the endpoint for the redox flows from food to final products, such as the cable bacteria that link individual cells into a chain so that one end can take up sulfur food sources and the other end connects to an oxygen source that provides the most efficient final processing of the food. Electrons flow along the chain in the metabolic cascade.

The book unifies many such variations in natural history and biochemistry. Frank’s ultimate argument is that to understand how natural processes have shaped such variety, one must form and test comparative hypotheses: predictions for how a change in the conditions is expected to alter the specific design in the biochemical processing of metabolism.

For example, greater competition between different microbial genotypes or species favors faster growth and less efficient yield. That outcome might be mediated by leakier cytochromes in electron transport, which increases metabolic throughput but reduces proton motive force and the total production of ATPs per unit of food taken up. Or greater fluctuation of food availability may favor enhanced food storage processes to survive periods of famine. The tuning of such storage versus growth depends on the intensity of competition for resources and the demographic opportunities for dispersal.

Overall, the book provides the first comprehensive framework linking microbial natural history to metabolic mechanism through testable evolutionary predictions. The approach provides a model for how to understand the natural processes that shape the design of organisms.

The paradox of robustness

In Maladaptation and the paradox of robustness in evolution¹¹⁷, Frank argued that robustness is inevitably coupled with decay. Herbert Spencer made the same point when he said “The ultimate result of shielding men from the effects of folly, is to fill the world with fools.” In other words, when robustness protects a system from the consequences of perturbations, the direct pressure of natural selection on the protected components weakens. Less selective pressure leads to evolutionary decay.

Frank used the example of cancer in Genetic variation in cancer predisposition: mutational decay of a robust genetic control network¹⁰². Our bodies have several processes that protect against the development of aggressive tumors. Some mechanisms limit cell division. Others kill cells with genetic damage or abnormal functions. Those mechanisms robustly protect against some types of mutation and cellular damage.

Suppose a new protective mechanism gets added over evolutionary time, such as enhanced immune recognition and control of early stage tumors. Initially, less cancer should occur. However, the prior mechanisms have less direct selective pressure acting on them, because if they fail there is now another protective mechanism to prevent disease. With the additional layer of robustness, the prior mechanisms may decay. Computer simulations show that such decay may arise by the increase in the frequency of inherited mutations in the protective mechanisms, raising the heritability of cancer.

Many aspects of physiology and regulatory control enhance homeostasis, which, by compensating for perturbations, is a form of robustness. In theory, as each homeostatic or robustness layer is added to a system, the existing components may decay a bit. Then, it becomes difficult to remove the new robustness layer, because the system has come to depend on it. Evolution proceeds by a ratchet dynamic, the layering of new robustness, relaxation of existing components, and irreversibility.

Frank developed these ideas initially in the context of cancer and regulatory or physiological systems, in which the protected components only partially decay because they are still strongly required functionally. In essence, each additional robustness layer alters the marginal costs and benefits of each component in the system, resetting the costs and the performance levels of the parts that work together to achieve function.

The theory of constructive neutral evolution was an earlier, independent idea that made a very similar argument^118–120. For example, RNA editing changes some of the nucleotides in an RNA transcript. If, in consequence, two distinct DNA nucleotides map to the same RNA nucleotide, then mutations that flip between the alternative DNA nucleotides become evolutionarily neutral, and drift occurs. Now the system cannot function if the RNA editing goes away, because the randomly changed DNA nucleotides would inevitably encode deleterious mutations. Here, the added layer of control in RNA editing renders a lower level component neutral. The logic is very similar to the paradox of robustness, but the emphasis is on neutral processes that make genetics more complex rather than on robust protection that alters the costs and benefits of components in a functional system.

Robustness and complexity, hourglasses and protocol waists

Frank’s article Robustness and complexity¹²¹ unified the paradox of robustness and constructive neutral evolution into a broad synthesis for how such evolutionary processes may influence the complexity of organisms. For example, each new process of robustness in gene expression leads to an irreversible layering of control. Over time, many controls will come to regulate expression, with a high level of complexity relative to what an engineer might design. Frank emphasized this point in Why are genomes overwired?¹²². In general, layered error-correcting controls occur throughout cell biology, physiology, behavior, and social systems.

Frank drew on John Doyle’s parallel work on robustness and complexity^123–125. In engineering and biology, complex systems often have to translate information from one domain to another. For example, computers must link software to hardware. The software does useful things. The hardware provides a generic basis for computation. The same software can typically run on any hardware because the software only talks to an intermediate protocol layer, the operating system.

The software may issue a generic command, such as “store these data.” The operating system protocols translate that command into the necessary message to send to the hardware.

Doyle’s insight is that essentially all complex robust systems have such protocols. In biology, DNA makes RNA makes protein is a universal translation between transmissible information in DNA and molecules that do things in proteins. Doyle emphasized that the protocol is a narrow waist between two broad halves of an hourglass or bowtie shape¹²⁶. For example, the transmissible information in genomes can diverge widely in mechanisms of storage transmission as long as it can still feed into the core protocol to translate into protein. Similarly, the proteins can diverge widely in function as long as they can still be built by the messages from the narrow cellular translation protocol, the narrow protocol waist that translates between the distinct information domains of nucleotides and proteins.

In the second part of Frank’s Robustness and complexity article, he argues that such protocol waists must be common throughout biological systems. For example, development passes through various narrow translations in protocol waists. Linking back to the paradox of robustness, we see that a protocol waist acts in part as a robustness layer, because it reduces the dimensionality of inputs to a few common messages. Such reduction makes the system invariant to many perturbations, which means that over evolutionary time, many variants have relatively small consequence and thus will easily diversify across species.

Doyle’s mechanistic hourglass becomes also an hourglass pattern of evolutionary diversity. Early and late processes in a system diverge evolutionarily, while the protocols that translate between them remain highly conserved between species. That matches the observed evolutionary hourglass of development^127–131. Frank argues that similar evolutionary hourglass patterns linked by conserved protocol waists are likely to occur in many biological systems.

Regulatory evolution, engineering principles in biology

The paradox of robustness is about how natural processes design the biochemical, physiological, and neural circuits that regulate biological processes. Frank explored other aspects of regulatory control in additional publications.

One line of study emphasizes control theory, which engineers use in the design of systems ranging from thermostats to the internet to advanced airplanes. Control theory has been widely applied to biological systems^123,132, but sometimes using only a limited or misleading interpretation of the theory.

For example, there is a result in control theory that says perfect adjustment to a changed setpoint requires integral control of errors¹³³. Integral control depends on the continuous summing of all error differences between the target point and the current circuit output. Many articles in biology cite this result and the claim that a particular circuit implements integral control, emphasizing the strong engineering prediction and the match to actual biology¹³⁴.

Although approximately true for biological circuits, Frank emphasized that biologists have misunderstood the natural simplicity of the problem and the misleading emphasis on the notion that biological circuits are designed to do complex integration^135,136. In practice, integral control is just a way of describing how error-correcting feedback must push strongly against errors to be effective, what is called high gain in control theory.

In biological circuits, high gain often arises simply and naturally by the balance between increases from the production of molecules and the decay of those molecules by breakdown or transformation to other products¹³⁶. So, it is approximately integral control, but Frank argued that saying so more simply and accurately would enhance biological understanding. A small misunderstanding, but one that illustrates how more sophisticated use of control theory concepts could enhance application to biology. Frank wrote his book Control Theory Tutorial: Basic Concepts Illustrated by Software Examples¹³⁵ to provide a better link between disciplines and to set the foundation for his own further studies.

Frank’s two articles on the Evolutionary design of regulatory control. I and II^136,137 provide the foundation for more sophisticated applications. The first article reviews the basic principles of control theory and then steps through the several key tradeoffs in control design as they apply to biochemical circuits. Several examples show how to derive specific predictions, for example, with regard to the tradeoff between better tracking goals and reduced system stability, and the tradeoff between plasticity-based adjustments to changed environments and the ability to maintain homeostatic setpoints.

The second article shows how biological circuits that have classic types of error-correcting control naturally accumulate particular types of genetic and phenotypic variability. In particular, as systems become more robust against perturbations, they can compensate for greater sloppiness in the performance of their components. That robust compensation reduces the force of natural selection on the system’s components, leading to component decay. This is the paradox of robustness, here analyzed within the details of a biochemical model that follows classic principles of robust control theory.

The analysis shows how increasingly robust systems accumulate more genetic variability and greater stochasticity of expression in their components. The theory predicts different levels of variability between different regulatory control architectures and different levels of variability between different components within a particular regulatory control system. The theory also shows that increasing robustness reduces the frequency of system failures associated with disease and, simultaneously, causes a strong increase in the heritability of disease. Thus, robust error correction in biological regulatory control may partly explain the puzzlingly high heritability of disease and, more generally, the surprisingly high heritability of fitness¹²¹.

A 2023 article Disease from opposing forces in regulatory control⁹⁷ provides another way to link the architecture of regulatory control to system failures that lead to disease. In this case, particular biological challenges cause control to be achieved by the balance between strongly opposing forces.

For example, in an earlier article with Paul Schmid-Hempel, Evolution of negative immune regulators⁹⁶, Frank noted that the immune responses against pathogen invasion often must be fast to be effective. That need for rapid triggering in response to a signal of attack means that the system will often produce costly false alarms. That logic is consistent with the fact that immune systems from plants to insects to vertebrates typically have a large group of negative regulators of immunity that function primarily to turn off immune responses. One prediction from that article is that strongly opposed positive and negative regulators increase the probability of misregulation and the heritability of immune-related disease.

Machine learning, evolutionary processes, and circuit design

Machine learning designs computational circuits to solve particular challenges. Natural selection designs biochemical and neural circuits to solve similar challenges. What can we understand about the architecture of biological circuits from the study of machine learning?

A key difficulty is that machine learning circuits typically have far more parameters than biological circuits can implement. Frank addressed this challenge in several articles, showing how optimized machine learning circuits can be reduced to simple architectures that predict the wiring of cellular circuits. He also explained the computational methods and showed how to transform them to study biological circuit design.

In Optimization of transcription factor genetic circuits¹³⁸, Frank modeled transcription factor (TF) circuits that control gene expression by using a combination of thermodynamic theory for TF binding and a common differential equation system for the dynamics of TFs and gene expression. The question is, how does natural selection tune the many rate parameters of this complex dynamic system to control gene expression in response to a given biological challenge.

Optimizing a complex system of differential equations has traditionally been a difficult challenge. Recent computational breakthroughs in machine learning provided a new method^139,140. Frank wrote a review of the method¹⁴¹ and an example application to ecological dynamics¹⁴².

Then, in the article mentioned above, he used the new method to analyze how a TF system could in principle entrain to external signals for a circadian rhythm in the face of erratic signals for entrainment and stochastic perturbations. The optimized system of stochastic differential equations achieved good performance, predicting a variety of architectural details in a dynamic TF circuit architecture that provide robustness to stochasticity and erratic input signals.

A subsequent article used the same approach to study the classic repressilator model of TF dynamics¹⁴³. The original repressilator model described a minimum deterministic differential equation system for TF dynamics that could maintain regular oscillations¹⁴⁴. The system oscillated regularly by having each TF repress the next TF in a cycle of three connected TFs. The parameters used in that original model did not arise from a broad search of alternatives but instead were chosen to produce regular oscillation rather than found through systematic optimization.

Frank’s new model extended the biological challenge from the existence of deterministic oscillations to the broader challenge of regular oscillations in the face of stochastic perturbations and irregular inputs for entrainment to an external signal. Computationally, this article improved Frank’s prior methods by using a combination of differential equations and neural networks to model the system, which provided a much more powerful computational search method.

The optimized circuit has the basic structure of the original repressilator but is enhanced with an extra TF and with architectural and rate details that significantly buffer internal stochasticity and extrinsic signal irregularity. The results provide predictions for the design of natural biological circuits and the engineering design of synthetic circuits, along with a method that can be widely applied to other challenges.

Another set of articles looked more generally at the kinds of challenges that are common to both machine learning and biology. Because we can study machine learning circuits simply and directly by computational analysis, the idea is that such computational study provides predictions about natural circuits.

In Precise traits from sloppy components: perception and the origin of phenotypic response¹⁴⁵, the challenge is how a system can evolve both perception and response to a signal when starting without either one. Perception without response has no value. Response without perception is not possible. In general, how do complex circuits get started when it is not obvious how an initial circuit can be at least partially functional?

Frank showed that a randomly connected network can provide the start. This application to biology follows from the machine learning method of reservoir computing^146–148. A randomly connected network takes inputs and flows random computations through the network nodes. By this process of computational flow, the network retains a trace of the input history, a form of perception that is encoded in a repeatable way although with no intrinsic meaning.

Then, the challenge is simply to match the particular nodal values to a beneficial response. In machine learning, one typically regresses the response on the nodal values, optimizing the regression to give a good response. That is computationally simple. Biologically it should also be easy, because the system only needs to evolve the regression coefficients to improve the response. One can end up with a relatively precise response that derives from random or sloppy internal components.

A pair of later articles, Circuit design in biology and machine learning. I and II,^149,150 provided a broader introduction to the use of machine learning as a model for biological circuit design, with many examples of how particular challenges may map between machine learning and biology. Examples include randomly connected networks, dimensional reduction to predict future trend in environmental inputs, and detecting environmental anomalies that signal danger.

The machine learning model for trend prediction discovered a classic method used to forecast future trends in the prices of financial securities¹⁵¹. By comparing a fast and slow moving average, a computational circuit predicts a rising trend when the fast average is above the slow average and a declining trend in the opposite case.

The computational model for dimensional reduction discovered exactly this calculation for trend prediction, achieving nearly optimal performance given the stochastically trending inputs used in the analysis. Interestingly, the circuit reduced to a very simple set of differential equations that could easily be calculated by a simple biochemical circuit, as summarized in the article A biological circuit to anticipate trend¹⁵².

The primary challenge in all of this arises from the fact that machine learning models typically have large numbers of parameters, making their circuits too complex to be implemented within the restricted biochemical constraints of cells. These articles address that challenge by showing how to reduce classic machine learning models to the small circuits that would predict the wiring of cellular circuits.

Natural selection, the Price equation, and theoretical unification

Natural selection forms a long thread through all of the work. How exactly does natural selection shape the design of organisms? What are the similarities and differences between natural selection and various learning or optimization processes? In the study of particular biological problems, better understanding of these questions is often needed to make progress.

Many of Frank’s individual advances and broader syntheses arise from the Price equation, a general mathematical expression that captures the essence of natural selection and learning processes^153–155. Frank’s early work showed how the Price equation elegantly subsumes Fisher’s fundamental theorem of natural selection, Robertson’s covariance theorem for quantitative genetics, the Lande–Arnold model for the causal analysis of selection, group and multilevel selection, and Hamilton’s rule for kin selection^9,12,57,156.

From 2011 to 2013, Frank published a series of seven articles on natural selection in the Journal of Evolutionary Biology, covering variable environments, developmental variability, levels of selection, the Price equation, information theory, path analysis, and the history of kin selection theory^{39,58,157–161}.

Later articles expanded the scope, linking the Price equation to broad aspects of statistical inference, Bayesian methods, information theory, thermodynamics, classical mechanics, and machine learning^162–167. A brief 2020 article Simple unity among the fundamental equations of science summarizes key results¹⁶⁸. Overall, Frank’s work has revealed the Price equation as a general tool for decomposing causes. Scholars across several disciplines have transferred Frank’s work to their own fields of study^169–174.

Frank’s recent preprint Fisher’s fundamental theory and regression in causal analysis¹⁷⁵ clarifies why the Price equation has been such a powerful tool. The article starts by showing that the Price equation is simply the exact product rule for finite differences, in which one splits the total change in the product of two variables into the change in the first variable holding the second constant plus the change in the second variable in the context of the changed value of the first variable. One can interpret this split as a component associated with the direct forces acting on a system plus a component associated with the changed context or frame of reference. Many scientific analyses arise from that split of total change^176–178.

A regression model links the values of predictor variables to predicted outcomes via the product of regression coefficients and predictor variables. A regression model can be interpreted purely as statistical prediction or as a model of hypothesized causes. In either case, the Price equation decomposition splits each regression term into the product of the change in the predictor variable between two contexts, holding the regression coefficient constant, plus the change in the regression coefficient in the context of the changed value of the predictor variable.

This split provides a series of partial all else equal terms that decompose total change into components of an overall prediction or causal interpretation. Science is, in essence, a search for the best all else equal decomposition of the forces that act on systems and the consequences of those forces for change^179,180. The Price equation is an exact expression of that decomposition and nothing more, thus providing the perfect tool for framing the scientific process.

Fisher’s fundamental theorem recognized that the regression coefficients perfectly describe the counterfactual concept of causality¹³. A regression coefficient is the effect that one obtains by changing each predictor one at a time from its initial to changed value and averaging those changes over the entire population of actual or imagined combinations of predictor variables. Thus, the regression is the perfect calculation of the particular counterfactual cause when evaluated in the specific current combinations of possibilities^180–182. The second term of the decomposition then augments change by accounting for how the altered context changes the regression coefficients.

Fisher discovered this decomposition in his study of the genetics of natural selection, calling it the fundamental theorem of natural selection when the predicted value is the mean fitness of a population. Although the theorem has not been particularly useful with regard to understanding fitness in biological populations, this counterfactual approach to causality has become the standard in modern causal analysis^180,182,183. Similar decompositions have been discovered independently in economics, demography, and thermodynamics^176–178. Again, the reason is that a split between direct forces and frame of reference, or a basic all else equal analysis, is the foundation for much of scientific analysis.

Finally, Frank’s 2025 paper in Entropy showed that a simple notational partitioning of change by the Price equation reveals what he termed a universal “force–metric–bias” law¹⁸⁴. Diverse learning algorithms, optimization methods, and natural selection all decompose into the same three components—a driving force, a metric that determines how the system responds to the force, and a bias term. This law unifies a wide variety of learning and statistical inference methods with natural selection into a single framework.

Common patterns, probability, symmetry, and scaling laws

Frank’s work on probability asks why common distributions, such as exponential, Gaussian, power law, and log series, recur across biological and physical systems whose underlying mechanisms differ enormously. The central claim is that recurrent patterns express the particular information, measurement structure, or symmetry that survives the aggregation of many hidden processes. Observed regularity therefore identifies an invariance class more often than a single causal story.

The early phase of this work was explicitly Jaynesian maximum entropy¹⁸⁵. In The common patterns of nature¹⁸⁶, Frank provided the central synthesis for this line of research. Neutral generative models are widely cited in biology, including scale-free networks¹⁸⁷, selectively neutral nucleotide substitutions¹⁸⁸, and neutral ecological dynamics¹⁸⁹. These neutral models matter not because they are privileged causal explanations, but because each preserves only a few informational constraints and therefore converges on a characteristic maximum-entropy form.

Frank’s key move was to treat those neutral models as clues to the many non-neutral processes that share the same constraints and generate the same patterns. Gaussian patterns arise when aggregation preserves only mean and variance, exponential patterns when only the mean is retained, and power laws when scale preserves the geometric mean. Many studies had linked biological patterns to maximum entropy. Frank¹⁸⁶ provided a broader synthesis of the underlying concepts and mathematics, and extended the scope and applicability of the framework to all commonly observed probability patterns.

With Eric Smith, Frank then made invariant measurement scales the main explanatory object in Measurement invariance, entropy, and probability¹⁹⁰ and A simple derivation and classification of common probability distributions based on information symmetry and measurement scale¹⁹¹.

These articles plus a companion on species abundance¹⁹² argued that information in measurement naturally changes with magnitude. It is the scaling of information that determines pattern and the associated form of the probability distribution. Essentially all common continuous probability distributions represent simple variants of the relationship between magnitude and information. In particular, different problems naturally grade between linear, logarithmic, or exponential scaling changes in information with magnitude. The seemingly unruly zoo of different common probability distributions becomes a simply ordered set of alternative scalings.

That logic provided a novel explanation for the Hill equation, a widely observed and puzzling pattern for how biochemical inputs transform into altered biochemical outputs. In Input-output relations in biological systems¹⁹³, Frank argued that the Hill equation’s prevalence reflects not the variety of specific biochemical mechanisms assumed by the standard theory¹⁹⁴ but the more general tendency of aggregation to dissipate microscopic detail and order responses into a small family of regular curves set by scale and information flow.

In 2014, Frank’s explanatory emphasis shifted from maximum entropy to symmetries or invariances. In Generative models versus underlying symmetries to explain biological pattern¹⁹⁵ and in the key synthesis How to read probability distributions as statements about process¹⁹⁶, Frank made the primary claim explicit. An observed probability pattern should usually be read not as evidence for a particular generative mechanism, but instead as a compact statement about what transformations leave the problem unchanged. Invariance is the primary explanatory principle. Maximum entropy emerges as a technique that is simply the consequence of underlying invariances. In application, the invariances define the range of potential causal processes that all attract to the same final observed form.

The primary emphasis on invariance became fully explicit in Common probability patterns arise from simple invariances¹⁹⁷. The work reorganized the subject around a small set of symmetries. Shift and stretch invariance yield the exponential-Boltzmann form, rotational invariance yields the Gaussian¹⁹⁸, and additional transformations and scaling relations carry these canonical patterns into the broader range of observed distributions. Companion papers applied that framework to the extreme value distributions and associated mortality and failure patterns¹⁹⁹ and to the ubiquitous power-law scaling of size distributions²⁰⁰.

Later applications showed the reach of the program. Frank’s 2018 PNAS article Measurement invariance explains the universal law of generalization for psychological perception²⁰¹ argued that Shepard’s exponential law of similarity follows from the minimal invariances that any sensible perceptual scale must satisfy. Ecological papers on the neutral theory of biodiversity with Jordi Bascompte²⁰² and on the log series and Zipf’s law²⁰³ showed that classic abundance patterns arise as simple expressions for the scaling of information within a unified invariance framework for probability patterns. The article on collective memory emphasized a recurring theme, that simple large-scale patterns are valuable because they identify invariance classes, but by themselves are not sufficient to determine mechanism²⁰⁴.

Across both the early Jaynesian maximum entropy work and the later symmetry-based program, Frank transformed commonly observed probability distributions from descriptive empirical facts into readable statements about information loss, measurement, and invariant structure.

The initial motivation for studying common patterns arose from Frank’s studies of cancer¹. Strikingly regular patterns of age-specific cancer incidence arise from widely varying mechanistic causes across tissues, exactly the kind of regularity that demands explanation beyond specific mechanism. Frank extended this line of reasoning to the age-specific mortality patterns of the leading causes of death in humans, which show similar overall regularity but informative variation between causes⁹⁸.

His 2009 synthesis The common patterns of nature¹⁸⁶ and 2016 Invariant death¹⁹⁹ linked these mortality and failure patterns to the family of extreme value distributions, connecting the abstract probability framework back to the concrete biological problems of cancer, aging, and death that motivated the work.

In both the early maximum entropy approach and the later symmetry-based program, Frank showed how to read empirical probability patterns in terms of measurement, invariance, and underlying cause. The links to cancer and mortality show how the abstract probability work plays an essential role in the fundamental challenge, understanding how natural processes shape biological design and how failure in disease reveals design.

References