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Title: Implementing Arithmetic and Other Analytic Operations By Transcriptional Regulation}
Article:
Abstract:
The transcriptional regulatory machinery of a gene can be viewed as a computational device, with transcription factor concentrations as inputs and expression level as the output. This view begs the question: what kinds of computations are possible? We show that different parameterizations of a simple chemical kinetic model of transcriptional regulation are able to approximate all four standard arithmetic operations: addition, subtraction, multiplication, and division, as well as various equality and inequality operations. This contrasts with other studies that emphasize logical or digital notions of computation in biological networks. We analyze the accuracy and precision of these approximations, showing that they depend on different sets of parameters, and are thus independently tunable. We demonstrate that networks of these “arithmetic” genes can be combined to accomplish yet more complicated computations by designing and simulating a network that detects statistically significant elevations in a time-varying signal. We also consider the much more general problem of approximating analytic functions, showing that this can be achieved by allowing multiple transcription factor binding sites on the promoter. These observations are important for the interpretation of naturally occurring networks and imply new possibilities for the design of synthetic networks.
Introduction:
In grappling with the biochemical complexity of gene regulation, some have turned to computational metaphors for explaining gene behavior( e. g.,[1]–[5]). The lac operon, for example, is often described as implementing a simple logical rule—the gene is on when lactose is present and glucose is absent[6]. Theoretically, a variety of nonlinear chemical systems, including gene regulatory systems, are capable of implementing arbitrary logical rules[7]–[11]. Networks of such systems can implement finite state machines[12], and families of such networks of increasing size can be said to implement arbitrary Turing-computable functions[13],[14]. In practice, logical models have proven capable of accounting for the qualitative dynamics of a variety of genetic systems[15]–[23]. Models that combine logical rules with concentration thresholds for the action of regulatory molecules, as in the French-flag model of Wolpert[24] or Glass networks more generally[25], satisfactorily describe other systems either qualitatively[26]–[28] or quantitatively[29]. Synthetic biologists have constructed gene networks that perform elementary logical operations, such as storing a bit of memory[30] or turning off and on with a fixed period[31]. However, detailed analysis of transcriptional regulatory networks reveals a behavior richer than logical responses. Yuh et al. 's model of the sea urchin developmental gene Endo-16 contains logical as well as additive and multiplicative operations[32]. Recent measurements of lac transcription rate as a function of the concentrations of cAMP and an analogue of allolactose show four plateaus of different rates connected by smooth boundaries[4],[5]. Even “logical” models of gene regulation often require more than two qualitatively distinct levels of gene expression( e. g.,[19]), recognizing that some genes cannot be treated simply as on or off. Besides logical or digital computation, several other notions of computation have been explored. Analog computations made by artificial neural networks can in principle be implemented by chemical systems[8],[12],[33]. Deckhard and Sauro experimented with evolving reaction networks to compute square and cubic roots of an input[34] and have since evolved networks to compute a variety of other arithmetic functions. This work shares the greatest commonality withours, the differences being that we specifically study single-gene transcriptional regulatory networks and that our designs are arrived at prescriptively, by analysis of the steady state equations, rather than by a computational search. A different line of reasoning has focussed on the robustness of biochemical networks to variability in inputs and parameters—intuitively important features for real systems[35]–[37]. Reduction in noise by development gene networks was experimentally observed[38],[39] and confirmed as a property of mathematical models[20],[40]. In this background, control theoretic concepts, especially noise filtering, were studied more carefully[41] and found in a wide variety of systems( see[37] for a summary). More recently, the behavior of several genes has been viewed in the context of information theory[42] and Bayesian decision theory[43]. The present work focuses on the general steady-state analog computational capacities of genes. Most of the paper considers a simple chemical kinetic model of transcriptional regulation for a single gene with two transcription factors. We assume that concentrations of proteins represent non-negative real numbers, with the transcription factor concentrations acting as inputs to the gene and the steady state expression of the gene acting as the output. Different choices for the kinetic rates allow the gene to approximate different binary arithmetic operations: addition, subtraction, multiplication, division, and testing for equality and various inequalities. Variations on the model allow alternative implementations of the same functions as well as other functions, such as the square root. In the model, these operations can be reproduced with arbitrary accuracy, although in reality, biological limits on the parameters would limit accuracy. We analyze the accuracy of these approximations in terms of the deviation between the steady state output and the desired output. We also analyze the precision of the approximations, in terms of the variability of the output over time in a stochastic interpretation of the model. Through theoretical analysis and simulations we show that such “arithmetic genes” can be combined in networks to compute more sophisticated functions. As an example, we describe an eight gene network that tracks the mean and standard deviation of a time-varying signal and detects times at which the signal is statistically significantly elevated. We also consider a model of a gene regulated by a single type of transcription factor but having multiple binding sites on the promoter. With such a gene, arbitrary analytic functions can be approximated up to a fixed order based on power series expansions. We demonstrate by designing a gene that approximates the cosine function.
Discussion:
We have shown that different parameterizations of a simple model of transcriptional regulation can reproduce binary arithmetic operations( addition, zero-truncated subtraction, multiplication, division), various inequality comparisons, and fractional powers. Unary, ternary, quaternary, etc. operations can be similarly implemented at a biochemical level. These models are not the only way, and may not be the best way, that these operations can be implemented. For example, we have not considered models with cooperativity between binding sites, allosteric features for transcription factors, regulation of translation or degradation, or chromatin-related mechanisms. One way to explore the utility of these mechanisms would be to perform an explicit computational search through some appropriate space of allowed reaction systems( as in Deckard and Sauro[34], for example), searching for systems with superior performance in terms of various metrics( such as accuracy, noise, robustness, evolvability, or energy consumption). Nevertheless, our simple models demonstrate the possibility of arithmetic computations at the level of transcriptional regulation. Other functions, such as exponentiation, logarithms, and trigonometric functions, could be computed by combinations of the arithmetic operations. However, this would be energetically wasteful and would require many genes. A more efficient approach is toapproximate directly by using more complicated regulatory machinery. We showed that a pair of genes, each with N binding sites for a transcription factor, can approximate any analytic function up to order N by reproducing its power series approximation. We also showed that by adjusting the parameters of such a gene pair, the analytic function can be approximated well over a larger range of inputs. While this drastically reduces the number of genes used to compute such functions, other mechanisms may be yet more efficient, and this is an avenue for future research. Our emphasis on arithmetic computations stands in contrast to work on logical or Boolean notions of computation in genetic networks. Interestingly, Hjelmfelt et al.[12] and Magnasco[13] both considered the problem of addition, but in the binary sense, with different chemical species effecting each bit of the operation. In general, there is no relationship between an arithmetic interpretation of a gene's behavior—which is accurate for a limited range of transcriptionfactor concentrations—and a logical interpretation—which focuses on the extremes of low and high transcription factor concentrations. For example, in our model for an Addition gene, transcription occurs whenever factor A or factor B is bound, keeping in mind that factors A and B cannot bind simultaneously( Figure 2A). Such a gene could be interpreted as implementing the logical OR function, because it would be expressed if either factor A or factor B is in high concentration. Recall, however, that an alternate design for an Addition gene allows A and B to bind simultaneously, as long as there was no transcription when both were bound. This gene would be interpreted as implementing XOR, as it would be expressed if A or B, but not both, were in high concentrations. Conversely, our designs for the zero-truncated subtraction function and the greater than function have identical logical structure( Figure 2A), but behave quite differently as arithmetic operations. Thus, a gene's arithmetic behavior, if any, is not determined by its logical behavior and vice-versa. This is an important caveat for the interpretation of empirical measurements of gene response surfaces, which may be probed at non-physiological concentrations of regulatory molecules. For our gene models to accurately approximate the desired operations, certain products of kinetic parameters must be equal. For example, the multiplication gene requires that rtzrabKoaKab = dtdz. Such a constraint may be biologically implausible or difficult to maintain over different cellular conditions. If the constraint did not hold, the expression of the multiplication gene would still be proportional to the product of the transcription factor expression levels—the role of the constraint is only to scale the output expression correctly. Thus, violation of the constraint in a particular situation need not invalidate the interpretation of the gene as performing multiplication. The case is similar for all the genes in Figure 2 except the addition gene. This gene requires rtzraKoa = rtzrbKob = dtdz. If those equalities failed, but raKoa = rbKob, then the expression of the gene would still be proportional to the sum of the transcription factor expression levels. If the latter equality failed as well, then the gene could be interpreted as performing a weighted sum. Indeed, we took advantage of this possibility in our design for the network that detects significant elevations in a time varying signal( see gene G7 in Figure 5A). In our models, precision( noise) in the output depends on the transcription factor binding and unbinding rates, while the accuracy of the output depends only on the ratio of the rates. This implies that there is no trade-off between precision and accuracy. Each can be selected independent of the other, by evolution or by synthetic biologists, subject to biological limitations on the parameters. We have not analyzed the speed with which these networks reach equilibrium, which could also be an important factor in some settings. Indeed, in many processes, such as development, expressing genes at the correct time is just as important as expressing them at the right level. Further, some genetic networks, such as circadian networks[52] or the feed-forward motif[50],[51], exist solely to keep time or process temporal signals. Our design and analysis of a network that tracks the first and second moments of a time varying signal and responds when the signal is statistically significantly elevated demonstrates that genes performing arithmetic computations can be useful in temporal information processing. An important avenue for further research is to consider more dynamical notions of computation, including investigation of how biochemical networks might implement computations from sequential signal processing or control theory. We expect that notions of computation more sophisticated than simple logical operations, already being documented experimentally[4],[5],[32],[53], will play an increasing role in our understanding of transcriptional regulation and of biochemical systems more generally. Our analysis of the analog computational abilities of transcriptional regulatory networks is a step in this direction.
The biochemistry of the cell is daunting in its complexity . In order to understand this complexity , we are often forced to use metaphors or construct analogies to systems that we understand better . One long-standing analogy is to digital computers , with their large networks of interacting components that manage to act in coherent and useful ways . Indeed , we know from both theoretical models and empirical observations that biological entities such as genes can sometimes be described accurately by digital , or logical , expressions—turning on or off in response to regulatory signals . However , far more sophisticated computations can also take place , as has been documented in the responses of genes such as the lac operon or Endo-16 . We analyze chemical kinetic models of transcriptional gene regulation and show that even simple models are capable of nearly arbitrary analog computations , ranging from elementary arithmetic operations to general analytic functions . Understanding the computational capacities of genes , and of biochemical systems more generally , tells us what to look for when studying natural systems and tells us what we can hope to build by biological engineering .