## 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.

This is a preview of subscription content, access via your institution.

## Notes

- 1.
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.

- 2.
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.

- 3.
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$$ \frac{\partial \theta}{\partial \alpha} = \frac{1}{2}\frac{\partial m(h(\alpha))}{ \partial \alpha} = \frac{1}{2}\frac{\partial h}{\partial \alpha} \chi(h(\alpha)) = \frac{1}{4 \alpha} \chi(h(\alpha)). $$ - 4.
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.

## References

- 1.
Hill TL (1985) Cooperativity theory in biochemistry. Springer series in molecular biology. Springer, Berlin

- 2.
Amit DJ (1987) Modeling brain function. Cambridge University Press, Cambridge

- 3.
Bialek W, Setayeshfar S (2005) Proc Natl Acad Sci USA 102:10040–10045

- 4.
Cook PF, Cleland WW (2007) Enzyme kinetics and mechanism. Garland Science, New York

- 5.
Marangoni A (2002) Enzyme kinetics: a modern approach. Wiley, London

- 6.
Karlin S (1980) J Theor Biol 87:33–54

- 7.
Thompson CJ (1972) Mathematical statistical mechanics. Princeton University Press, Princeton

- 8.
Karlin S, Kennet R (1980) Math Biosci 87:97–115

- 9.
Acerenza L, Mizraji E (1997) Biochim Biophys Acta 1339:155–166

- 10.
Wong J (1975) Kinetics of enzyme mechanics. Academic Press, New York

- 11.
Edsall JT, Wyman J (1958) Biophysical chemistry. Academic Press, New York

- 12.
Hill AV (1910) J Physiol 40:4–7

- 13.
Changeux JP (1961) Cold Spring Harb Symp Quant Biol 26:313–318

- 14.
Margelefsky EL, Zeidan RK, Davis ME (2008) Chem Soc Rev 37:1118–1126

- 15.
Cui Q, Karplus M (2008) Protein Sci 17:1295–1307

- 16.
Ellis RS (2005) Entropy, large deviations and statistical mechanics. Springer, Berlin

- 17.
Diamant H, Andelman D (2000) Phys Rev E 61:6740–6749

- 18.
Agliari E, Barra A, Camboni F (2008) J Stat Mech 10:10003–10020

- 19.
Agliari E, Barra A (2011) Europhys Lett 94:10002–10007

- 20.
Barra A (2008) J Stat Phys 132:787–809

- 21.
Monod J, Wyman J, Changeux JP (1965) J Mol Biol 12:88–118

- 22.
Voet D, Voet JG, Pratt CW (2008) Fundamentals of biochemistry: life at the molecular level. Wiley, London

- 23.
Murase Y, Onda T, Tsujii K, Tanaka T (1999) Macromolecules 32:8589–8594

- 24.
Chatterjee AP, Gupta-Bhaya P (1996) Mol Phys 89:1173–1179

- 25.
Barra A (2009) Math Methods Appl Sci 32:783–797

- 26.
Cases JM, Villieras F (1992) Langmuir 8:1251–1254

- 27.
Pagnani A, Parisi G, Ricci-Tersenghi F (2000) Phys Rev Lett 84:2026–2029

- 28.
Burioni R, Cassi D, Blumen A (2002) Chem Phys 282:409–417

- 29.
Agliari E, Barra A, Camboni F (2011) Rep Math Phys 68:1–22

## Acknowledgments

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

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

### Keywords

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