MultivalentInteractions

Interactions between molecules, proteins, vesicles, and cells are hardly the result of only one bond formation but instead, they involve a multitude of bonds that act holistically to create unique association profiles. Such a cooperative nature is paramount to control interaction not only in their magnitude but most importantly in their selectivity. When it comes to biological organisation and life in general, the actors are numerous, with each unit surrounded by thousands of different other units each characterised by their unique chemical signature. Yet each interaction is finely controlled with an extremely high level of precision and selectivity. How do cells, virus, vesicles, proteins, achieve such a level of precision? How do they distinguish one from the other?

These are the questions we are now asking and using statistical physics and biophysical techniques we are now studying multivalent interaction in natural systems. We also re-engineer them into synthetic surrogates to create a new therapeutic tool we call phenotypic medicine.   

 

(E1)

Molecular interaction and its control is the bread and butter of medicinal chemistry. Understanding them allows us not only to understand their implication in terms of biochemical and biophysical processes but it enables us to make drugs to control these processes. At the beginning of the 20th century, Nobel laureate Paul Ehrlich classified molecules into ligands and receptors and their interaction selectivity depends on their chemical affinity toward each other. 

Such an interaction can be idealised as an associative reaction between the ligand and the receptor to form a ligand/receptor complex. The association constant of such a reaction can be used to estimate the binding energy according to equation (E1).

A multivalent systems with, l, ligands binds to a surface with r, receptors, forming λ bonds which maximun is the min(r,l). The association constant is not merely the sum of the single binding events, but it becomes holistic. A multivalent system binds to a multivalent surface according to a two-step process. The first interaction is the same as the monovalent, and one ligand binds to its receptor exposing the other ligands to form other bonds. The association constant of each further bond depends on interaction constant KI that encapsulates the internal equilibrium between single and multi bonded states. As a first approximation, we can assume that KI is proportional to the monovalent bond kA and the binding volume as in E2, with NA being the Avogadro constant. With multivalent systems, we never get one single ligand/complex but an ensemble of Ωtot arrangements. With Ωtot defined as the degeneracy coefficient, we can now write the total binding energy (E3) as the combination of three elements; (i) the monovalent binding, (ii) the free energy associated with multiple bonds, and (iii) a purely entropic term, known as avidity entropy, due to the increase of degeneracy.

 

(E2)

(E3)

An alternative approach to defining the binding between the two multivalent units is to derive the partition function at the bound state. For multivalent system, the partition function can be derived as in E4 across all possible λ bonds. Each of them releasing free energy βEbinding where β=kT  is the product between the Boltzmann constant, k, and the temperature, T. Such a statistical physics approach allows us to derive the total binding energy as in E5 and most importantly, we can break interaction down to the single partition functions. 

(E5)

(E4)

TheTheory

 
 
 
 

Sterics

(E1)

Molecular interaction and its control is the bread and butter of medicinal chemistry. Understanding them allows us not only to understand their implication in terms of biochemical and biophysical processes but it enables us to make drugs to control these processes. At the beginning of the 20th century, Nobel laureate Paul Ehrlich classified molecules into ligands and receptors and their interaction selectivity depends on their chemical affinity toward each other. 

Such an interaction can be idealised as an associative reaction between the ligand and the receptor to form a ligand/receptor complex. The association constant of such a reaction can be used to estimate the binding energy according to equation (E1).

A multivalent systems with, l, ligands binds to a surface with r, receptors, forming λ bonds which maximun is the min(r,l). The association constant is not merely the sum of the single binding events, but it becomes holistic. A multivalent system binds to a multivalent surface according to a two-step process. The first interaction is the same as the monovalent, and one ligand binds to its receptor exposing the other ligands to form other bonds. The association constant of each further bond depends on interaction constant KI that encapsulates the internal equilibrium between single and multi bonded states. As a first approximation, we can assume that KI is proportional to the monovalent bond kA and the binding volume as in E2, with NA being the Avogadro constant. With multivalent systems, we never get one single ligand/complex but an ensemble of Ωtot arrangements. With Ωtot defined as the degeneracy coefficient, we can now write the total binding energy (E3) as the combination of three elements; (i) the monovalent binding, (ii) the free energy associated with multiple bonds, and (iii) a purely entropic term, known as avidity entropy, due to the increase of degeneracy.

 

(E2)

(E3)

An alternative approach to defining the binding between the two multivalent units is to derive the partition function at the bound state. For multivalent system, the partition function can be derived as in E4 across all possible λ bonds. Each of them releasing free energy βEbinding where β=kT  is the product between the Boltzmann constant, k, and the temperature, T. Such a statistical physics approach allows us to derive the total binding energy as in E5 and most importantly, we can break interaction down to the single partition functions. 

(E5)

(E4)

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