21.8: Specificity in Recognition and Binding

Specificity in Recognition

What determines the ability for a protein to recognize a specific target amongst many partners? To start, let’s run a simple calculation. Take the case that a protein (transcription factor) has to recognize a string of n sequential nucleotides among a total of N bases in a dsDNA.

• Assume that each of the four bases (ATGC) is present with equal probability among the N bases, and that there are no enthalpic differences for binding to a particular base.
• Also, the recognition of a particular base is independent of the other bases in the sequence. (In practice this is a poor assumption).
• The probability of finding a particular n nucleotide sequence amongst all n nucleotide strings is

\[\left( \dfrac{1}{4} \right)^n \]

• For a particular n nucleotide sequence to be unique among a random sequence of N bases, we need

\[\left( \dfrac{1}{4} \right)^n \geq \dfrac{1}{N} \]

• Therefore we can say

\[ n \geq \dfrac{\ln{N}}{\ln{4}} \]

Example

For the case that you want to define a unique binding site among N = 65k base pairs:

• A sequence of n = ln (65000)/ln(4) ≈ 8 base pairs should statistically guarantee a unique binding site.
• n = 9 → 262 kbp

This example illustrates that simple statistical considerations and the diversity of base combinations can provide a certain level of specificity in binding, but that other considerations are important for high fidelity binding. These considerations include the energetics of binding, the presence of multiple binding motifs for a base, and base-sequence specific binding motifs.

Energetics of Binding

We also need to think about the strength of interaction. Let’s assume that the transcription factor has a nonspecific binding interaction with DNA that is weak, but a strong interaction for the target sequence. We quantify these through:

∆G₁: nonspecific binding

∆G₂: specific binding

Next, let’s consider the degeneracy of possible binding sites:

g_n: number of nonspecific binding sites = (N – n) or since N ≫ n: (N – n) ≈ N

g_s: number of sites that define the specific interaction: n

The probability of having a binding partner bound to a nonspecific sequence is

\[\begin{aligned}
P_{\text {nonsp }} &=\frac{g_{n} e^{-\Delta G_{1} / k T}}{g_{n} e^{-\Delta G_{1} / k T}+g_{s} e^{-\Delta G_{2} / k T}} \\ &=\frac{(N-n) e^{-\Delta G_{1} / k T}}{(N-n) e^{-\Delta G_{1} / k T}+n e^{-\Delta G_{2} / k T}} \\ &=\frac{1}{1+\frac{n}{N} e^{-\Delta G / k T}} \end{aligned}\]

where ∆G = ∆G₂ – ∆G₁. We do not want to have a high probability of nonspecific binding, so let’s minimize P_nonsp. Solving for ΔG, and recognizing P_nonsp 1,

\[ \Delta G \leq -k_BT\ln{\left[ \dfrac{N}{nP_{nonsp}} \right] }\]

Suppose we want to have a probability of nonspecific binding to any region of DNA that is P_nonsp ≤ 1%. For N = 10⁶ and n = 10, we find

\[ \Delta G \approx -16k_BT \qquad \textrm{or} \qquad -1.6 k_BT/nucleotide \]

for the probability that the partner being specifically bound with P_sp > 99%.

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Readings

1. G. Schreiber, G. Haran and H. X. Zhou, Fundamental aspects of protein−protein association kinetics, Chem. Rev. 109 (3), 839-860 (2009).
2. D. Shoup, G. Lipari and A. Szabo, Diffusion-controlled bimolecular reaction rates. The effect of rotational diffusion and orientation constraints, Biophys. J. 36 (3), 697-714 (1981).
3. D. Shoup and A. Szabo, Role of diffusion in ligand binding to macromolecules and cell-bound receptors, Biophys. J. 40 (1), 33-39 (1982).