A biochemical logarithmic sensor with broad dynamic range

Sensory perception often scales logarithmically with the input level. Similarly, the output response of biochemical systems sometimes scales logarithmically with the input signal that drives the system. How biochemical systems achieve logarithmic sensing remains an open puzzle. This article shows how a biochemical logarithmic sensor can be constructed from the most basic principles of chemical reactions. Assuming that reactions follow the classic Michaelis-Menten kinetics of mass action or the more generalized and commonly observed Hill equation response, the summed output of several simple reactions with different sensitivities to the input will often give an aggregate output response that logarithmically transforms the input. The logarithmic response is robust to stochastic fluctuations in parameter values. This model emphasizes the simplicity and robustness by which aggregate chemical circuits composed of sloppy components can achieve precise response characteristics. Both natural and synthetic designs gain from the power of this aggregate approach.


Introduction
I present a simple biochemical circuit that logarithmically transforms input signals. This circuit adds the outputs of several reactions that follow standard mass action Michaelis-Menten kinetics. Alternatively, the biochemical kinetics may follow the commonly observed Hill equation response, which includes Michaelis-Menten kinetics as a special case. This sensor has high dynamic range, responding logarithmically across many orders of magnitude. The high dynamic range is achieved by adding together reactions with different sensitivity ranges. The aggregate nature of this circuit provides robustness to parameter variations. Aggregate sensor design may explain the commonly observed high dynamic range of logarithmic biological responses and may also provide a useful tool for synthetic biology.

Results and discussion
Many biochemical reactions and cellular responses transform an input, x, into an output, y, according to the Hill equation in which c is the value of the input x that yields one-half of the maximal output response, k is the Hill coefficient that determines the shape of the response, and y is normalized to be between 0 and 1 1 . Simple mass action kinetics often follows the Hill equation with k = 1, which corresponds to classical Michaelis-Menten kinetics 2 . For example, the input may drive production of the output, and the output may spontaneously transform back to a prior state.
The output of the Hill equation scales approximately logarithmically with its input through the middle part of its response range, because y is roughly linear with respect to log x. Prior studies have emphasized that a Hill equation response can act as a logarithmic sensor 3,4 . However, a single Hill equation response provides a logarithmic sensor with limited dynamic range ( Figure 1A).

Amendments from Version 2
This version adds two new figures to show how the logarithmic sensor responds for various parameter combinations.

REVISED
My extended dynamic range sensor arises by adding together n Hill equations with increasing values of the half-maximal response, c i , as For example, if c i = b i , then each reaction in the sum has an increasing input value at which its maximal sensitivity occurs. Figure 1B-E shows that simple combinations of k and b create a logarithmic sensor, in which the output is linearly related to the logarithm of the input. The logarithmic relation holds robustly when the parameters k and b of the individual reactions vary stochastically ( Figure 1F). Figure 2 and Figure 3 show the response for various parameter combinations.
Several biochemical circuit responses and many aspects of perception scale logarithmically 5 . A robust and generic pattern of this sort seems likely to depend on a robust and generic underlying design. In the search for a generic circuit, my biochemical logarithmic sensor has three advantages over prior designs. First, prior models depended on particular molecular assumptions about biochemical kinetics or reaction pathways 3,4 . My design requires only Michaelis-Menten or Hill equation responses, which are very widely observed in biochemical and cellular systems 1 . Second, prior models focused on single input-output processes,  which have relatively narrow dynamic range. My aggregate design provides a logarithmic response over a vastly greater range. Third, the prior models' responses are easily perturbed by parameter fluctuations. My design performs robustly with respect to broad fluctuations in parameters. The robustness of my logarithmic sensor emphasizes the potential to achieve precise response characteristics from underlying sloppy components when using an aggregate design 6,7 .

Competing interests
No competing interests were disclosed.

Grant information
National Science Foundation grant DEB-1251035 supports my research.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. This is a brilliant short paper: a clever idea, explained very clearly and that immediately makes sense (but that is by no means obvious at first sight). The revision and the comments about the first version seem appropriate to me. I don't have further substantial comments except perhaps the following.

Open Peer Review
I was curious to understand what value of is required to achieve the result presented in the figure (is n there anything special about =7?). Using the code provided this is easy to check, but perhaps it would n help readers to add an additional figure with multiple curves for values of ranging between 1 and 7 to n see how the results change. Doing this exercise myself I learned, for instance, that <7 is enough for n certain parameters and that the value of is especially important (something that perhaps could be b discussed in a sentence or two).

If applicable, is the statistical analysis and its interpretation appropriate? Not applicable
Are all the source data underlying the results available to ensure full reproducibility? Yes

Are the conclusions drawn adequately supported by the results? Yes
No competing interests were disclosed.

Competing Interests:
I have read this submission. I believe that I have an appropriate level of expertise to confirm that 1. The author presents an interesting model of a robust biochemical logarithmic switch with a broad dynamical range as a sum of multiple Hill-equation based models.

I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
My main problem with this work is the lack of a biological context with a real example (the author argues the model applies to natural as well as artificial circuits). Is the author proposing that the successive Hill-type switches would be arranged in series? A more natural example would be a layered cascade like a MAP kinase cascade . But in that case the output would not be a simple linear sum as in the author's Eqn. (1), but rather something like a Hill function of a Hill function. 1 1.
layered cascade like a MAP kinase cascade . But in that case the output would not be a simple linear sum as in the author's Eqn. (1), but rather something like a Hill function of a Hill function.
The contention that "My design performs (more) robustly with respect to broad fluctuations in parameters" should be established more rigorously. How were the stochastic simulations performed? It's hard to tell since no methodological details are provided.
Second, I have uploaded the source code, as a supplementary file, that I used to do the stochastic simulations. The short, simple code provides a clear, exact description of the methods. The code was used to generate Figure 1, and can be used to regenerate that figure and explore alternative assumptions. A summary of the methods are provided in the legend for Figure 1.
The final point concerns the lack of specific biological examples and details. The first aspect of this comment is a question asked by Dr. Bhattacharya: "Is the author proposing that the successive Hill-type switches would be arranged in series?" I am proposing a summation of individual responses to a stimulus, as described in Equation 1. For example, suppose that a set of sensors respond to sound intensity. Each sensor has a Hill-like response curve, but tuned to different intensity level. The output of the aggregate set of sensors is the sum of the individual sensor responses. Each sensor may comprise a simple or complex set of reactions. All that we care about here is that the net response per sensor is Hill-like in shape.
That same logic could describe the response to a molecular signal. Various reactions could each have a Hill equation response, but tuned to different sensitivities. The aggregate consequence of those responses would be a net output level.
With regard to an explicit example, I gave a lot of thought to that issue while preparing the manuscript. I could not find a compelling example for which sufficient biological detail about mechanism had been described. There are plausible cases that are worth exploring, such as the sensory problems that I mentioned. For molecular systems, there are hormones for which cells have multiple receptors, presumably tuned differently with regard to response and other aspects (e.g., estrogen receptors). However, I agree that those multireceptor systems do not provide a compelling case given current knowledge.
Why are there no compelling molecular examples? I think the reason is that researchers rarely look for responses that arise by aggregate processes, such as summation of underlying reactions. Instead, the search is almost always for a single reaction or explicit pathway that exactly matches the desired input-output pattern. So my goal for this article is to suggest a plausible alternative for consideration.
One advantage of publishing with F1000Research is that I can update the article in the future. If a good example arises in discussion with readers of the article or from the literature, I will update the article to include that example. For now, I think it is most important that the novel idea be published in a concise and direct manner, with emphasis on the simple logic of the model.
No competing interests were disclosed. Competing Interests: