Signal integration and information transfer in an allosterically regulated network

A biological reaction network may serve multiple purposes, processing more than one input and impacting downstream processes via more than one output. These networks operate in a dynamic cellular environment in which the levels of network components may change within cells and across cells. Recent evidence suggests that protein concentration variability could explain cell fate decisions. However, systems with multiple inputs, multiple outputs, and changing input concentrations have not been studied in detail due to their complexity. Here, we take a systems biochemistry approach, combining physiochemical modeling and information theory, to investigate how cyclooxygenase-2 (COX-2) processes simultaneous input signals within a complex interaction network. We find that changes in input levels affect the amount of information transmitted by the network, as does the correlation between those inputs. This, and the allosteric regulation of COX-2 by its substrates, allows it to act as a signal integrator that is most sensitive to changes in relative input levels.

example of how multiple inputs lead to multiple outputs in a physiological context. We focus our study 48 on the allosteric regulation network of COX-2 by two important substrates, arachidonic acid (AA) and

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In this work, we analyze the execution mechanism of a biochemical reaction network with multiple 54 inputs. Our work explains how a MIMO system integrates information on the concentration and nature 55 of its substrates to yield potentially different outputs. In previous work, we developed a detailed model   93 We first explored the net flow of reaction flux through the network using a graph theoretic approach to 94 calculate all possible paths between the unbound enzyme and each final product. Briefly, we evaluated 95 the system of ordinary differential equations (ODEs) in CORM at time intervals to extract the integrated 96 reaction flux at a given time point for each chemical reaction. We then built paths from product to reactant 97 following the reactions with net forward flux. Finally, we calculated the total chemical flux that passed 98 through a given path and used this as a measure of the probability of product formation via that path; 99 a detailed description of this procedure is given in SI Methods and Fig. S1. All fluxes were calculated 100 for the first ten seconds of catalysis after mixture with the substrates, a time chosen to match previous 101 experimental work (Mitchener et al., 2015). Path flux distributions were calculated for an ensemble of 102 calibrated parameter values to quantify path flux uncertainty arising from parameter uncertainty.

Substrate-Dependent Reaction Fluxes in Signal Execution
Our analysis indicates that there are six possible paths to produce PG (Fig. S2) and four possible paths to produce PG-G (Fig. S3)   production, we also found that the flux through each dominant path for PG-G production is dependent on 115 substrate concentration (Fig. 2).

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In the absence of 2-AG and at low (0.5 µM) AA, PG is produced without allosteric modulation ( Fig.   117 2A, purple; purple-labeled path in Figure 1C, top); as the concentration of AA increases, the proportion of 118 PG produced with AA as an allosteric modulator also increases ( Fig. 2A, green). When 2-AG is added 119 to the system, PG production shifts to using 2-AG as an allosteric modulator ( Fig. 2A, red), with this 120 path favored to a greater extent as the concentration of 2-AG increases ( Fig. 2A, lower plots). Even in amount of 2-AG (0.5 µM) is added to the system, more than half of PG production occurs via a 2-AG or 123 AA allosterically modulated path. In the presence of high concentrations of either modulator, as much as 124 90% of PG is produced via an allosterically modulated path.  We note that at any given substrate concentration, the uncertainty arising from the calibrated kinetic 138 parameter distributions never exceeds a 20% change in the percentage of product produced by a given 139 path ( Fig. S4-S8). We find that changes in substrate levels and their relative ratios have a much larger 140 effect on the dominant reaction paths than changes in kinetic rates within the calibrated CORM parameter 141 distributions. Overall, these findings suggest that variation of substrate concentrations in physiologically-142 relevant ranges has a significant impact on COX-2's mechanism of catalysis.

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Calculating the flux through each path allows us to obtain information about the preferred sequences of reactions that the system executes while processing AA and 2-AG. However, these measurements do not provide an estimate of how chemical traffic (i.e. the flow of chemical signals in the network) is distributed throughout the network. To explore the distribution of biochemical network traffic, we introduce the pathway entropy to quantify the degree to which COX-2 utilizes multiple paths at different concentrations of substrates. Our definition of entropy, originally introduced by Claude Shannon (Shannon, 1948) provides a measure of the uncertainty in a probability distribution across states as follows: where H is entropy and P(x i ) is the probability of any state x i . To determine the degree of uncertainty 145 associated with product production (the pathway entropy), we considered each pathway as a state and use 146 the fraction of flux that a given pathway contributes to the product as a measure for the probability of 147 that state. This analysis yields a measure of how evenly distributed production is across possible paths.

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In general, evenly distributed fluxes across paths in a network would maximize pathway entropy for a 149 multi-path system.

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Since the dominant paths vary with substrate concentration (Fig. 2), we would expect that pathway 151 entropy would also vary. In Fig. 3 we present the pathway entropy dependence on input concentration for 152 PG (Fig. 3A) and PG-G (Fig. 3B). The pathway entropy for PG production is highest at intermediate levels top plot, center). In contrast, in the lowest entropy states -low AA and high 2-AG for PG ( Fig. 2A  the suppression of PG-G formation is greater than that of AA formation as seen in the lower plateau level 180 achieved in (Fig. 4B). In addition, the range of inputs over which the output varies depends significantly on 181 which input/output pair is chosen (note the difference in that range in Fig. 4A,B). Clearly, variation of both

Channel Capacity from Substrates to Products
To better understand how this output variation, combined with the shape of the concentration-dependence curves, influences the COX-2 reaction network, we applied an additional concept from information theory to measure dependence between inputs and outputs, namely the Mutual Information: where X represents a given signal and Y the response to that signal (Shannon, 1948 has focused on estimating the "channel capacity," which is the maximum information attainable across all possible input distributions: Note that any practical calculation provides a lower bound estimate for the channel capacity C, since only

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single parameter set is shown in Fig. 4. Greater detail is provided in the SI Methods.

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When the inputs are varied independently, channel capacity values decrease even further (Fig. 5B   228 and C). The channel capacity between AA and PG or PG-G is generally less than 2 bits, and the channel 229 capacity between 2-AG and those outputs is generally less than 1.5 bits. This could occur for two reasons. consistently present in a 2-to-1 AA-to-2-AG ratio ( Fig. S10 and Fig. S11). The behavior when input 244 ratios were fixed was similar to that for the correlated values (when the input levels were fixed equal to 245 each other); channel capacities were again higher than in the independent case and the effect of kinetic 246 parameter variation on channel capacity was higher. When the inputs are moderately correlated, the 247 system is still able to obtain high channel capacities for some kinetic parameter sets, although the overall 248 distribution of channel capacities shifts to lower values compared to when input correlation is perfect, 249 further confirming COX-2 input integration.

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We binned the input data into four quadrants (high or low values of either substrate) and calculated the 253 channel capacity between inputs and outputs independently for each quadrant; input ranges were otherwise 254 identical to those used for the calculations described above. Low substrate values spanned 0-8 µM and 255 high substrate values 8-16µM. Both independently varied inputs (Fig. 6A) and correlated inputs (Fig. 6B) 256 yielded estimated channel capacities that were significantly different between the different regions of 257 input space. In addition to differences in PG and PG-G channel capacity, we found that the distribution of 258 information that passed through different intermediates changed with substrate concentration (Fig. S13   259 and Fig. S14); certain paths to product had greater information transfer capacity at particular levels of 260 substrates. This echoes findings from our pathway analysis (Figs. 2 and 3), indicating that changes in 261 substrate concentration result in significant changes in how the enzyme executes its catalytic mechanism.

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Interestingly, we found no detectable correlation between the flux through a pathway and the mutual 263 information between an input and an intermediate in that path ( Fig. S15 and Fig. S16). We leave further 264 investigation of the relationship between information transfer and actual physical reaction fluxes for future 265 work.

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Splitting the input space into different quadrants also revealed signficant variation between different 267 parameter sets, with most distributions showing significant bimodality across parameters (Fig. 6). This 268 suggests that both the shape of the concentration-dependence curves, and the impact of "extrinsic noise"

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COX-2 has significant regulatory flexibility: it is an allosteric protein, with multiple substrates and 293 multiple allosteric regulators, all of which can influence how COX-2 operates on its substrates in vivo.

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The pathway analysis ( Fig. 1B and 1C) suggests that COX-2 functions by first binding a substrate at 295 the catalytic site, followed by binding of an allosteric regulator. Allostery can be viewed as a shift these conformations are more easily turned over to product than the unmodulated enzyme, while for