# BE.180:SecondOrderBinding

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### Second Order Binding (of two things)

Givens:

• A physical interaction between molecules A and B.
• A system that contains some initial concentration of molecules A and B ($\displaystyle A_0$ and $\displaystyle B_0$ , respectively).

Tasks:

• Compute the steady state concentrations of free A, free B, and the A:B complex.

Approach:

• Write differential equation for change in A:B over time.
$\displaystyle \frac{d[A:B]}{dt}=+k_{on}*[A]*[B]-k_{off}*[A:B]$
• Solve equation at steady state (that is, no change in concentration of the A:B complex.
$\displaystyle 0=+k_{on}*[A]*[B]-k_{off}*[A:B]$
• Solve for $\displaystyle K_D$ , the dissociation constant.
Equation 1: $\displaystyle K_D = k_{off}/k_{on} = \frac{[A][B]}{[A:B]}$
• Note constraints on system due to conservation of mass.
Equation 2: $\displaystyle [A_0] = [A] + [A:B]$
Equation 3: $\displaystyle [B_0] = [B] + [A:B]$
• Note system of three unknowns with three equations (1-3 above)! Solve for unknowns A, B, and A:B (takes you through a quadratic).