Models in biology: ‘accurate descriptions of our pathetic thinking’

In this essay I will sketch some ideas for how to think about models in biology. I will begin by trying to dispel the myth that quantitative modeling is somehow foreign to biology. I will then point out the distinction between forward and reverse modeling and focus thereafter on the former. Instead of going into mathematical technicalities about different varieties of models, I will focus on their logical structure, in terms of assumptions and conclusions. A model is a logical machine for deducing the latter from the former. If the model is correct, then, if you believe its assumptions, you must, as a matter of logic, also believe its conclusions. This leads to consideration of the assumptions underlying models. If these are based on fundamental physical laws, then it may be reasonable to treat the model as ‘predictive’, in the sense that it is not subject to falsification and we can rely on its conclusions. However, at the molecular level, models are more often derived from phenomenology and guesswork. In this case, the model is a test of its assumptions and must be falsifiable. I will discuss three models from this perspective, each of which yields biological insights, and this will lead to some guidelines for prospective model builders

The rational parameterisation theorem for multisite post-translational modification systems

Post-translational modification of proteins plays a central role in cellular regulation but its study has been hampered by the exponential increase in substrate modification forms (“modforms”) with increasing numbers of sites. We consider here biochemical networks arising from post-translational modification under mass-action kinetics, allowing for multiple substrates, having different types of modification (phosphorylation, methylation, acetylation, etc.) on multiple sites, acted upon by multiple forward and reverse enzymes (in total number L), using general enzymatic mechanisms. These assumptions are substantially more general than in previous studies. We show that the steady-state modform concentrations constitute an algebraic variety that can be parameterised by rational functions of the L free enzyme concentrations, with coefficients which are rational functions of the rate constants. The parameterisation allows steady states to be calculated by solving L algebraic equations, a dramatic reduction compared to simulating an exponentially large number of differential equations. This complexity collapse enables analysis in contexts that were previously intractable and leads to biological predictions that we review. Our results lay a foundation for the systems biology of post-translational modification and suggest deeper connections between biochemical networks and algebraic geometry

Beware the tail that wags the dog: informal and formal models in biology

ABSTRACT Informal models have always been used in biology to guide thinking and devise experiments. In recent years, formal mathematical models have also been widely introduced. It is sometimes suggested that formal models are inherently superior to informal ones and that biology should develop along the lines of physics or economics by replacing the latter with the former. Here I suggest to the contrary that progress in biology requires a better integration of the formal with the informal. In a series of previous essays, I discussed how formal mathematical models have played a far more significant role in biology than most biologists typically appreciate The word "model" has many meanings in biology. We speak of model organisms as institutionalized representatives of particular phyla. We occasionally build physical models, as Crick and Watson did for DNA. Mostly, however, a model refers to some form of symbolic representation of our assumptions about reality, and that is the sense in which I will use the word here. An informal model is one in which the symbols are mental, verbal, or pictorial, perhaps a scrawl of blobs and arrows on the whiteboard; in contrast, a formal model is one in which the symbols are mathematical. Informal models pervade biology. They help to guide our thinking, and experimentalists rely on them to design experiments. The model may turn out to be nonsense, and an experiment may reveal that, but one has to start somewhere. It is sometimes claimed that one starts with data, from which a model is constructed. But why those data? And how should those data be interpreted? The answers reveal informal models that precede the acquisition of data. Models, whether informal or formal, allow us to capture assumptions and to undertake reasoning. Informal models have two classes of assumptions: those that are explicit in the model itself, or foreground assumptions; and those that are only implicit but potentially significant, or background assumptions. In molecular biology, a foreground assumption might be that blob X is an activated enzyme, which implements an informal arrow. A background assumption might be that X has multiple posttranslational modifications, which influence activation but differ depending on the organism. Whether a particular fact is in the foreground or relegated to the background depends on the problem at hand and the questions being asked. This allows us to tolerate much ambiguity. Does X mean chicken X or fl

Thermodynamic bounds on ultrasensitivity in covalent switching

Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch, in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants, and thereby reflect fundamental physical constraints on covalent switching.Comment: 29 pages, 6 figure

A Complex Hierarchy of Avoidance Behaviors in a Single-Cell Eukaryote.

Complex behavior is associated with animals with nervous systems, but decision-making and learning also occur in non-neural organisms [1], including singly nucleated cells [2-5] and multi-nucleate synctia [6-8]. Ciliates are single-cell eukaryotes, widely dispersed in aquatic habitats [9], with an extensive behavioral repertoire [10-13]. In 1906, Herbert Spencer Jennings [14, 15] described in the sessile ciliate Stentor roeseli a hierarchy of responses to repeated stimulation, which are among the most complex behaviors reported for a singly nucleated cell [16, 17]. These results attracted widespread interest [18, 19] and exert continuing fascination [7, 20-22] but were discredited during the behaviorist orthodoxy by claims of non-reproducibility [23]. These claims were based on experiments with the motile ciliate Stentor coeruleus. We acquired and maintained the correct organism in laboratory culture and used micromanipulation and video microscopy to confirm Jennings' observations. Despite significant individual variation, not addressed by Jennings, S. roeseli exhibits avoidance behaviors in a characteristic hierarchy of bending, ciliary alteration, contractions, and detachment, which is distinct from habituation or conditioning. Remarkably, the choice of contraction versus detachment is consistent with a fair coin toss. Such behavioral complexity may have had an evolutionary advantage in protist ecosystems, and the ciliate cortex may have provided mechanisms for implementing such behavior prior to the emergence of multicellularity. Our work resurrects Jennings' pioneering insights and adds to the list of exceptional features, including regeneration [24], genome rearrangement [25], codon reassignment [26], and cortical inheritance [27], for which the ciliate clade is renowned.DBT-Cambridge Lectureshi

The rational parameterisation theorem for multisite post-translational modification systems

Post-translational modification of proteins plays a central role in cellular regulation but its study has been hampered by the exponential increase in substrate modification forms (“modforms”) with increasing numbers of sites. We consider here biochemical networks arising from post-translational modification under mass-action kinetics, allowing for multiple substrates, having different types of modification (phosphorylation, methylation, acetylation, etc.) on multiple sites, acted upon by multiple forward and reverse enzymes (in total number L), using general enzymatic mechanisms. These assumptions are substantially more general than in previous studies. We show that the steady-state modform concentrations constitute an algebraic variety that can be parameterised by rational functions of the L free enzyme concentrations, with coefficients which are rational functions of the rate constants. The parameterisation allows steady states to be calculated by solving L algebraic equations, a dramatic reduction compared to simulating an exponentially large number of differential equations. This complexity collapse enables analysis in contexts that were previously intractable and leads to biological predictions that we review. Our results lay a foundation for the systems biology of post-translational modification and suggest deeper connections between biochemical networks and algebraic geometry

Unlimited multistability in multisite phosphorylation systems

Reversible phosphorylation on serine, threonine and tyrosine is the most widely studied posttranslational modification of proteins (1, 2). The number of phosphorylated sites on a protein (n) shows a significant increase from prokaryotes, with n less than or equal to 7 sites, to eukaryotes, with examples having n greater than or equal to 150 sites (3). Multisite phosphorylation has many roles (4, 5) and site conservation indicates that increasing numbers of sites cannot be due merely to promiscuous phosphorylation. A substrate with n sites has an exponential number (2^n) of phospho-forms and individual phospho-forms may have distinct biological effects (6, 7). The distribution of these phospho-forms and how this distribution is regulated have remained unknown. Here we show that, when kinase and phosphatase act in opposition on a multisite substrate, the system can exhibit distinct stable phospho-form distributions at steady state and that the maximum number of such distributions increases with n. Whereas some stable distributions are focused on a single phospho-form, others are more diffuse, giving the phospho-proteome the potential to behave as a fluid regulatory network able to encode information and flexibly respond to varying demands. Such plasticity may underlie complex information processing in eukaryotic cells (8) and suggests a functional advantage in having many sites. Our results follow from the unusual geometry of the steady-state phospho-form concentrations, which we show to constitute a rational algebraic curve, irrespective of n. We thereby reduce the complexity of calculating steady states from simulating 3 times 2^n differential equations to solving two algebraic equations, while treating parameters symbolically. We anticipate that these methods can be extended to systems with multiple substrates and multiple enzymes catalysing different modifications, as found in posttranslational modification 'codes' (9) such as the histone code (10, 11). Whereas simulations struggle with exponentially increasing molecular complexity, mathematical methods of the kind developed here can provide a new language in which to articulate the principles of cellular information processing (12)