2.6 Subsite Theory
2.6.1 Linear polymer
The Subsite Theory [28] postulates that the active site of an enzyme consists of several subsites, which are responsible for binding the substrate, and one catalytic site which is responsible for hydrolyzing the bond. This is consistent with visualization of 3D structures of enzymes [212]. Each monomer unit can only bind to one subsite and each subsite can only bind one monomer unit. This binding is reversible.
Definitions and assumptions
For ease of discussion, the subsites are numbered (note that other numberings are sometimes used in literature): The first subsite gets the number 1. There are z˘ subsites and the catalytic center lies between subsitez˘−andz˘−+1. The number of subsites above the catalytic center definesz˘+. Further-more, the monomer units of the polymer are indexed with the reducing end having the number 1 and the non-reducing end assigned to the numberk˘0. s˘is defined as the index of the monomer unit which occupies the subsite 1.
The polymer has to occupy contiguous subsites. This means that, if subsite 3 and 5 are occupied, subsite 4 needs to be occupied. Furthermore, the polymer is oriented in such a way that the index of the monomer unit decreases with the subsite number. This means that if monomer unit 7 occupies subsite 3 then monomer unit 8 occupies subsite 2 and not 4. Additionally, the polymer cannot leave the active site of the enzyme. This means that, if monomer unit 2 occupies subsite 3, subsite 2 can only be empty if the polymer has only two monomer units [28].
Reaction mechanism
All this assumptions together allow the formulation of the reaction mechanism:
E + Ss,˘˘k0 ESs,˘k˘0 E + S˘k0−˘k + Sk˘ k+1,˘s,k˘0
k−1,˘s,k˘0
k+2,s,˘k˘0
For each polymer many bindings are possible. Precisely, the binding modes s˘∈[1,z˘+k˘0−1] are possible. Furthermore, because the bond at the catalytic center is hydrolyzed one can specify k˘ =
˘
s−z˘−−1. It follows trivially that the catalytic constants is only different from zero if the catalytic center is occupied. This is the case ifs˘∈[˘z−1+1,z˘−+k˘−1].
Hydrolysis rate
The reaction mechanism describes a reaction of Michaelis-Menten type. Accordingly, each binding mode can be described with the following equation, if quasi-steady state is assumed and the enzyme concentration is sufficiently low
Ds,˘k˘0=(k+2,s,˘k˘0·Is,˘k˘0) E•·n(˘ k)˘ 1+n(˘ k)˘ ·P˘k0+z−1˘
i=1 Ii,k˘0
(2.55)
Ii,k˘0= k+1,s,˘k˘0
k−1,s,˘k˘0+k+2,s,˘k˘0
. (2.56)
This equation for each mode is quite inconvenient and one desires one equation for each amount of monomer unit. By summing up the hydrolysis over all productive modes one obtains this equation which is of Michaelis-Menten type:
D(k˘0) =s˘cat(k˘0)·n(˘ k)˘ ·E•
1+n(˘ k)˘ ·I˘(k˘0) (2.57)
I˘(k˘0) =
k˘0+˘z−1
X
˘ s=1
I˘s,˘k˘0 (2.58)
˘
scat(k˘0) =kcat(k˘0)·I(˘k˘0) =
k˘0+˘z−−1
X
˘ s=˘z−+1
k+2,s,˘˘k0·Is,˘k˘0. (2.59)
Microscopic constants
In the Subsite Theory it is assumed that the subsites do not further interact with each other. The association constantK˘0
˘
s,k˘, which describes how strongly a polymer withk˘0 monomer units binds to the
enzyme in positions˘, is then given by
where the cratic free energy of mixing∆Gmixingis the contribution of the solvent to the binding energy
∆Gmixing=R·T·ln 55.51, (2.61)
wherex=55.51 is a physical constant of water [215, ch.5]. The unitary free energy of binding∆Gi is the contribution of the binding of the polymer to subsiteito the binding energy. This contribution is zero for imaginary subsites (i<0ori>z˘++z˘−). Under the assumption that the hydrolysis of the bond is much slower than the binding of the polymer (assumption of fast equilibrium) the Michealis constant for each binding modeKm,s,˘k˘0 is equal to the inverse of the association constant for this binding mode. An alternative way of stating this is to say that the inverse of the Michaelis constant for each binding mode I˘s,˘k˘0 is equal to the association constant for this binding modeK˘0
˘ s,˘k0
K˘s,0˘k˘0 ≈I˘s,˘k˘0. (2.62) In the standard Subsite Theoryk+2,s,˘˘k0 would be only temperature dependent and the equation above would be fully specified. However, some authors [125, 216] have found that the Subsite Theory cannot describe the hydrolysis by all enzymes. They have introduced anad hoc strain to allow for interaction between subsites. It is assumed that the binding of a substrate monomer unit lowers the activation-energy barrier by a constant amount. This constant amount∆Gais called the acceleration factor.
k˘+2,s,˘˘k0 =
The proportionality factorCis a product of the temperature dependent catalytic constant and the time and temperature dependent concentration of active enzyme. Both of which can be described using the Equilibrium Model. If the acceleration factor is set to zero, the standard Subsite Theory is obtained again. Nevertheless, the specificity of the enzyme can be computed.
Product distribution function
If the polymer of lengthk˘0 is hydrolyzed while in positionk˘+z˘−+1, two polymers one withk˘ monomer units and another withk˘0−k˘ monomer units are produced. The polymer of lengthk˘ is also produced from positionk˘0−k˘+z˘−+1. By summing up the production rate due to this two positions and division by the total production rate, the PDF is obtained
γ(˘ k,˘ k˘0) = 1
Simplification for sufficiently large polymer
For polymers with more monomer units than there are subsites one can simplify the equations because the binding modes that occupy all bindings sites have the same affinity K˘int0 and the same catalytic constantk˘+2,max
If the polymer has more than2·max(z˘−,z˘+)monomer units, the product distribution can be expressed as a weighted sum of ECS and random scission on inner monomers
˘
The estimation of the binding energies ∆Gi, the acceleration factor∆Ga, and the proportionality con-stantC between catalytic constant and the reaction rate was studied by Allen and Thoma [216] who found that given the likelihood of cleavage of a bond, Michealis constants, and the maximal reaction rates the parameters could be well estimated. The likelihood of cleavage of a bond can be obtained by letting the enzyme cleave a radioactive labeled polymer of a certain chain length for a short time followed by chromatography separation of the products [217]. The Michaelis constants and catalytic rates are obtained by letting the enzymes act for a short time on a polymer of a certain chain length and measuring the increase in polymer concentration. By performing repeated experiments at different con-centration one can then apply the standard Lineweaver-Burk method to obtain the Michealis constant and the maximal reaction rate from the initial reaction rate [217].