## Abstract

We consider cooperative reactions and we study the effects of the interaction strength among the system components on the reaction rate, hence realizing a connection between microscopic and macroscopic observables. Our approach is based on statistical mechanics models and it is developed analytically via mean-field techniques. First of all, we show that, when the coupling strength is set positive, a cooperative behavior naturally emerges from the model; in particular, by means of various cooperative measures previously introduced, we highlight how the degree of cooperativity depends on the interaction strength among components. Furthermore, we introduce a criterion to discriminate between weak and strong cooperativity, based on a measure of “susceptibility.” We also properly extend the model in order to account for multiple attachments phenomena: this is realized by incorporating within the model *p*-body interactions, whose non-trivial cooperative capability is investigated too.

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## Notes

In chemical kinetics, one is typically interested in finding the rate law governing a reaction, i.e., how the concentration of a given reactant or product varies in time. For elementary reactions (i.e., reactions with a single mechanistic step) the law of mass action is valid, and the reaction rate is proportional to reactant concentrations raised to a power defined by stoichiometric coefficients, but this is no longer true for complex reactions. However, if the substrate molecules react on binding sites to form a product, as is the case of enzymes, we expect the rate for product formation to be still proportional to the fraction of occupied sites θ.

We notice that the entropic term is connected to the logarithm of the number of configurations associated with a given fraction θ of occupied sites: It is maximized for fractions around 1/2, which have a larger number of configurations associated, and minimized for θ = 0, θ = 1, corresponding to just one configuration and a vanishing entropy.

Note that in the frame of the Curie–Weiss model this is strictly related to the generalized susceptibility

$$ \chi = \frac{\partial m(h)}{\partial h} $$which measures the response of the system to a change in the field

*h*. In fact, we have$$ \frac{\partial \theta}{\partial \alpha} = \frac{1}{2}\frac{\partial m(h(\alpha))}{ \partial \alpha} = \frac{1}{2}\frac{\partial h}{\partial \alpha} \chi(h(\alpha)) = \frac{1}{4 \alpha} \chi(h(\alpha)). $$When there is not such a symmetry, one should more properly consider the quantity

$$ n_H = 4 \left. \frac{\partial \theta}{\partial \log \alpha}\right|_{{\rm max, min}}. $$(21)that is the slope of θ(log α) calculated at the inflection point, where it has an extremum. In fact for an unsymmetrical system the slope at θ = 1/2 is not generally an extremum slope and should not be taken as a suitable cooperativity index.

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## Acknowledgments

The research belongs to the strategy of exploration funded by the FIRB project RBFR08EKEV which is acknowledged.

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Di Biasio, A., Agliari, E., Barra, A. *et al.* Mean-field cooperativity in chemical kinetics.
*Theor Chem Acc* **131**, 1104 (2012). https://doi.org/10.1007/s00214-012-1104-3

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DOI: https://doi.org/10.1007/s00214-012-1104-3

### Keywords

- Michaelis–Menten
- Hill
- Binding isotherm
- Statistical mechanics
- Ising model
- Reaction-kinetics