3.7.3 Shift
In the case of the final step where we calculate the effective concentration by integrating the product of the receptor and ligand PDFs, we can state that the PDF of the ligand and the receptor starts from the same origin, the center of the binding site. On the other hand, we can introduce shifts at the starting point to model, for example, the reverse states, where the hinges of the linkers fall to opposite ends of the receptor-ligand complexes.
In this cases the form changes from:
Ef fnum = CL·CRR R R
R(r)·L(r)r2sin(ϕ)drdϕdθ CU
to in case where there is no shift:
Ef fnum = CL·CR·4πR R(r)·L(r)r2dr
CU (3.24)
or to in case of when there is a shift:
Ef fnum = CL·CRR R R R(r)·L(√
shif t2+r2 −2shif t·rcosϕ)r2sin(ϕ)drdϕdθ
CU (3.25)
Also, it could be beneficial to rewrite the spherical coordinates to cylindrical coordinates resulting in a much simpler equation:
Ef fnum = CL·CR·2πR RR(√
r2+z2)·L(qr2+ (z+shif t)2·r dr dz
CU (3.26)
3.8 Different Effective Concentrations within Multivalent binding
In bivalent binding we can only differentiate two distinctCef f effective concentrations. To derive it we can start from equation 2.8 in an extended form between receptor R∅,∅ and ligand L1,2. Then using the notation of equation: dx(t)dt = A(t)x(t), then if the x(t)′ is [[R∅∅],[R1∅L1,2],[R2∅L1,2],[R∅,1L1,2],[R∅,2L1,2],[R1,2L1,2],[R2,1L1,2]than the correspond- ing state matrixAis:
−k0−1on [L1,2]
−k0−2on [L1,2]
−k0−3on [L1,2]
−k01−4on [L1,2]
+kof f1−0 +kof f2−0 +k3−0of f +kof f4−0
+k0−1on [L1,2] +kof f1−0 −kon1−5
+k0−2on [L1,2] +kof f2−0 −k2−6on
+k0−3on [L1,2] +k3−0of f −k4−6on
+k0−4on [L1,2] +kof f4−0 −kon4−5 +kon1−5 +kon4−5
−kof f5−1
−kof f5−4
+kon2−6 +k3−6on
−k6−2of f
−k6−3of f
(3.27)
If the binding capabilities of the two lignad domain is the same towards the two receptor domains thankon0−1 =kon0−2 =k0−3on =kon0−4 =Konas well askof f1−0 =k2−0of f =kof f3−0 =kof f4−0 = kof f. Furthermore if the bivalent binding does not affect the dissociation thank1−0of f =k5−1of f = k5−4of f =kof f6−2 =kof f6−3 =kof f.
Also knowing the integral of the PDFR f(R|r0) = R f(r0|R)therefore the Cef f1−5 = Cef f4−5 and this concentration represent that the numbering of the ligand and receptors are inline or in a straight conformation and this effective concentration isCef fs . Also Cef f2−6 = Cef f3−6 concentration represent that the numbering of the ligand and receptors are in reverse order or in a twisted conformation and this effective concentration isCef fr . Using these substitution than the state matrixAsimplifies to:
−4kon[L1,2] +kof f +kof f +kof f +kof f
+kon[L1,2] +kof f −konCef fs
+kon[L1,2] +kof f −konCef fr
+kon[L1,2] +kof f −konCef fr
+kon[L1,2] +kof f −konCef fs
+konCef fs +konCef fs −2kof f
+konCef fr +konCef fr −2kof f
(3.28)
3.8. DIFFERENT EFFECTIVE CONCENTRATIONS WITHIN MULTIVALENT BINDING 55
3.8.1 Effective Concentration Notation
In trivalent cases, there are many more different effective concentrations. Therefore, if the spacing of the ligand or receptor units is consistent (which is the most scenario due to the repeating DNA segments in biology) can be generalized. We categorize the effective concen- trations based on the position of multivalent binding; if two neighboring partners participate in the binding on the receptor, then we can talk about type "1" binding. Also, if there is one unit spate between the two participating units, it is type "2" binding and so on. This also apply for the Ligand therefore any binding event can be categorised like ligand type "1" and receptor type "2" which gets a notationL1 2ef f to differentiate if from the effective concentra- tions which has different notation therefore, the bivalentCef fs =L1 1ef f andCef fr =L−1 1ef f . To get the type of the binding, we can subtract the position of the new binding from the position of the already bound unit. If the receptor and ligand both have positive or negative signs, the binding is inline type; otherwise, it is twisted type binding.
Furthermore, starting from trivalent-trivalent binding, there are cases where both neighbors of the unit are already bound, and it forces the position of the middle unit. This type of binding has notation X; we can also note the distance from the preceding and the distance from the following unit in the binding resulting in notation likeX1 1.
3.8.2 Comparing Effective concentration Models
To compare the effect of described effective concentration calculation, we can compare it with the most straightforward model [25, 66] with randomly distributed ligand and receptor ends. In this model, one molecule is divided by the volume that the receptor and ligand occupy; this approximation of the effective concentration can give similar results in the case of highly flexible longer linkers, but in other cases, this difference can be significant see Figure 3.2.
In the simple model, the ligand and receptor ends are randomly distributed, resulting in a uniform PDF integrated over the volume of possible binding. Therefore the difference between the two scenarios is the largest in cases where the distribution is skewed, like in the case of a relatively rigid linker. In Figure 3.2 B, four cases are presented with the detailed and simple model. Presented cases can have flexible or rigid linkers as well as cases when the ligand and receptor linkers have the same length; therefore, the units are in-register, and in cases where the linkers have different lengths, therefore the units are out-of-register.
The cases are: trivalent receptor and ligand with (i) flexible, in-register; (iii) flexible, out-of- register; (v) rigid, in-register; or (vii) rigid, out-of-register receptors and ligands.
Figure 3.2: Effective ligand concentrationsCef f, determined with a uniform model in which ligand is uniformly distributed in a volume with a radius of the linker contour length fails to describe the positional steric effects of both rigid and out-of-register receptor-ligand linkers that are reflected in the linker-driven PDF used in our zero-fit model. (A) The configurational ensemble for a trivalent receptor-trivalent ligand interaction evolves via 12 types of intra- complex association, each specified with an individual [Leff] that determines the first-order rate constant of association. (B) Comparison of the linker-driven PDF calculation used in our model with a non-structured, uniform concentration calculation ofCef f. Cef f calculations are shown for both models using four of the simulated trivalent interactions. Linkages within the trivalent receptor and ligand are (i) flexible, in-register; (iii) flexible, out-of-register; (v) rigid, in-register; or (vii) rigid, out-of-register. [43]