The authors have declared that no competing interests exist.
Conceived and designed the experiments: EB. Performed the experiments: EB AC CD LH SH SM MM AN JW. Analyzed the data: EB SH. Wrote the paper: EB.
A significant proportion of enzymes display cooperativity in binding ligand molecules, and such effects have an important impact on metabolic regulation. This is easiest to understand in the case of positive cooperativity. Sharp responses to changes in metabolite concentrations can allow organisms to better respond to environmental changes and maintain metabolic homeostasis. However, despite the fact that negative cooperativity is almost as common as positive, it has been harder to imagine what advantages it provides. Here we use computational models to explore the utility of negative cooperativity in one particular context: that of an inhibitor binding to an enzyme. We identify several factors which may contribute, and show that acting together they can make negative cooperativity advantageous.
Most enzymes operate in multi-subunit complexes, and a significant proportion, perhaps 25% or more, display cooperativity in binding ligand molecules
Here we consider the impact of cooperativity on metabolic regulation in the case of inhibitor binding to an enzyme. In this situation binding at one subunit can influence the affinity of other subunits for the same inhibitor, thereby affecting how the rate of catalysis responds to inhibitor concentration. In
End-product inhibition provides a natural context for examining cooperativity in metabolic regulation, and a number of studies have touched on its impact in this case
The value of negative cooperativity in metabolic regulation has received less attention. However this represents an important question because negative cooperativity is almost as common as positive among enzymes in nature
Consider the metabolic network in
A. The network consists of two molecules connected by one reaction. The reaction is inhibited by the product molecule 1. There are fixed rate outflow reactions from molecule 1. These take on different levels in different environments, as illustrated by the arrows. B. Plot of the activity of the enzyme vs. concentration of the inhibitor (molecule 1). The two curves have half-saturation at molecule 1's target value, and illustrate the shape with two different Hill coefficients. C. A plot showing how fitness varies with Hill coefficient for this network, given that the half-saturation is set to molecule 1's target. Fitness (see text for definition) reflects the deviation of molecule 1 from its target and thus is in units of concentration.
We explore this computationally using simple kinetic models of enzymatic reactions. In the network in
In the absence of regulation a new steady state will be reached via mass action. In the case of a lowered outflow, the concentration of molecule 1 will begin to build up. The rate of the back reaction (1
Regulation can allow the system to achieve a new steady state while minimizing the deviation of molecule concentrations from their target values. Regulation in our system is achieved through the action of one or more non-competitive inhibitors on an enzyme. We model inhibitor binding with the Hill equation which has been shown to provide a good fit to ligand binding data in a wide variety of situations
A regulated solution to the network in
A systematic examination of the relationship between metabolic homeostasis and Hill coefficient in this network is consistent with our expectation. For a given Hill value we can calculate the deviation of molecule 1 from its target in all environments. We define fitness as the negative of the average of the absolute values of these deviations. The larger the fitness, the better the homeostasis. We examined fitness for Hill coefficients from 0.25 to 3.98 for this simple network, and the results are shown in
We next consider a variation on this situation where the maximum possible Hill does not give the fittest solution. Like our first example, the network shown in
A. Illustration of the network. Molecules 3 and 4 have constant rate outflows which do not vary in different environments, shown with black arrows. B. A fitness landscape for this network. The x and y axes give Hill coefficient for inhibition of the x and y reactions. The z axis is fitness, which reflects the deviations of molecules from their targets and is in units of concentration (see text for definition). C. Plot of the activity of the enzyme vs. concentration of the inhibitor (molecule 2).
We are once again interested in finding a way to regulate this network to maintain metabolic homeostasis. We now have multiple molecules and we will consider homeostasis to involve minimizing the deviation of all of them from their targets. Thus fitness will be defined as follows. We determine the deviations of all molecules from their targets, considered across all environments. The negative of the average of the absolute values of these deviations is the fitness.
The relationship between Hill coefficients and fitness is shown by the fitness landscape in
To understand why it is advantageous to have the reaction x Hill less than the maximum, consider the plot in
We looked for this pheonomenon over a range of conditions, varying enzyme amount and the magnitude of the baseline outflow over an order of magnitude. We find that for a wide range of parameter values, the best Hill value is less than the maximum possible. The size of the effect is dependent on the magnitude of the net flow through reaction x relative to the magnitude of the back reaction through x. The effect is most robust if the flow is greater than or equal to about
Lets next consider what happens when we extend the simple pathway in
A. Pathways with 2, 3 and 4 molecules. B. Plots of fitness vs. Hill coefficient of inhibitor binding given homeostasis on all molecules. C. Fitness vs. Hill coefficient given homeostasis acting only on the tip molecule. Fitness reflects the deviations of molecules from their targets and is in units of concentration (see text for definition).
We again used an evolutionary approach to search for the optimal combination of Hill coefficients and half-saturations for these networks. For our example set of parameters (see
These longer networks are very similar to what was considered by Hofmeyr and Cornish-Bowden
The reason for the drop in optimal Hill for longer pathways is that there is a delay in concentrations equilibrating in the long branch. Let us consider what happens in the 4 molecule network when we test with the lower outflow environment. Imagine that the Hill coefficient is at the maximum possible value. The inhibitor concentration (molecule 3 in this case) begins to increase. We slide down the inhibition curve, and at some point inhibition causes the rate of flow through the inhibited reaction to equal the new rate of outflow from the product molecule. But the system is not yet at steady state because the two internal reactions do not yet have the same rate as the new outflow. The first reaction can reach that level quickly through inhibition. But the other two will need to do so through mass action, which takes longer. Their equilibration will cause the concentration of molecule 3 to continue to increase, leading to overinhibition, and the system eventually settling in a steady state that undershoots the final target concentrations. In this situation, fitness can be optimized by having a less steep inhibition curve which does not constrict entry into the pathway as quickly.
We systematically explored parameter space for this system, to try to understand the conditions under which the optimal Hill coefficient drops when we go from two to three molecules in our pathway. Using the range of Hill coefficients used above (0.25–3.98), we found that roughly 20% of parameter combinations produced this effect. In another 50% of the cases the optimal Hill for both two and three was at the maximum of 3.98, but if we allowed ourselves to go to Hills above this value we found that the effect was still present (i.e optimal Hill for three was less than that for two).
Because this effect depends on the gradient of concentrations that is established in a pathway with flow going through it, the relative energies of the molecules in that pathway will impact it. We found that running our examples in energetically unfavorable pathways tends to make our effect stronger (i.e. less-than-maximum Hill coefficients are more useful in longer pathways), and running them in energetically favorable pathways tends to reduce the magnitude of the the effect.
Our goal is to identify advantages of negative cooperativity in metabolic systems. So far we have described two phenomena which result in less-than-maximum Hill values, but not negative cooperativity. Using relatively extreme parameter values (e.g. a branching network with 12 branches) we found that we could get negative cooperativity to be advantageous. However we would like to find less extreme cases where this is so. We next discuss a small network which exhibits negative cooperativity in inhibitor binding (
B. A fitness landscape for this network. The x and y axes give Hill coefficient for inhibition of the x and y reactions. The z axis is fitness, which reflects the deviations of molecules from their targets and is in units of concentration (see text for definition).
We again used an evolutionary approach to explore the optimal solutions for this network. In this case, we decided to not only vary the Hill and half-saturation on reactions x and y, but to also allow the identity of their inhibitors to vary. This made the parameter space we explored larger, but also gives us more confidence that we are finding the global optimum solution for the network.
We did 32 independent runs, and the best solution in each of these followed the same general pattern. They all involved molecule 4 inhibiting both reaction x and y, half-saturations for both reactions set around molecule 4's target value, and negative cooperativity (Hill
reaction x Hill | 0.34–0.51 |
reaction y Hill | 1.59–3.32 |
reaction x half-saturation | |
reaction y half-saturation |
Range of Hills and half-saturations for best organisms from 32 runs on the network in
This network provides an illustration of how negative cooperativity can arise in a metabolic network. It combines the two phenomena we have discussed above. First it is a branching network with nested end-product inhibition. Second the pathway producing this end-product has several intermediates. Our results show that acting together, these two phenomena are capable of producing negative cooperativity, at least under certain sets of parameters.
Before concluding, we would like to address one objection which could be made to our approach. We have used the Hill equation to model inhibitor binding on the basis that it provides a good empirical fit to ligand binding data
We have suggested two factors to explain the evolution of negative cooperativity in metabolic regulation. First, in branching networks with nested end-product inhibition, it is advantageous for the inhibitor to bind the enzyme before the branch with a smaller Hill coefficient than it binds the one after. This is because the branch point enzyme needs to be modulated less strongly than enzymes after the branch. If both are inhibited by the same end product, then it is optimal for the branch point enzyme to have a less sharp response to inhibitor. The second factor occurs in linear, unbranching pathways with end-product inhibition. It is due to a difference in the time scale between inhibition on the one hand, and mass action on the other. An important condition for this second factor is that metabolic homeostasis be acting on intermediates as well as products of the pathway, a situation which is likely to be common in nature
We have used simple models to demonstrate that these two factors occur over a range of parameter values, and that in combination they can lead to negative cooperativity. A natural extension would be identify pathways which are known to exhibit end product inhibition with negative cooperativity in nature
Our theoretical results are a step toward understanding the role of negative cooperativity in metabolic regulation. Beyond its intrinsic interest, this has the potential to aid biotechnology as humans seek to design and modify metabolic pathways for our own purposes.
Our numerical simulations of chemical reactions were written in python using the scipy package's odeint function. Our code can be downloaded from our website at
We used a two-step reversible Michaelis-Menten mechanism for our reactions:
This gives us the following rate equations:
No inhibitor:
Here
We obtained the various rate constants (e.g.
The network in
We systematically explored the space of parameters for several of our networks (
Our evolutionary simulations involved mutation and selection without recombination. The simulation for
Thanks to Karl Haushalter, Arend Hintze, Daryl Yong, and Lisette de Pillis for discussions.