The protein SmpB binds to a special species of tRNA, tmRNA, to eliminate the incomplete proteins made from truncated mRNAs in bacteria. If the binding of SmpB to tmRNA is plotted as fraction tmRNA bound versus SmpB concentration, one obtains a symmetrical S-shaped curve as shown in Figure Q3–3. This curve is a visual display of a very useful relationship between K_d and concentration, which has broad applicability. The general expression for fraction of ligand bound is derived from the equation for K_d (K_d = [Pr][L]/[Pr–L]) by substituting ([L]_TOT – [L]) for [Pr–L] and rearranging. Because the total concentration of ligand ([L]_TOT) is equal to the free ligand ([L]) plus bound ligand ([Pr–L]), fraction bound = [Pr–L]/[L]_TOT = [Pr]/([Pr] + K_d)

Figure Q3-3

For SmpB and tmRNA, the fraction bound = [SmpB– tmRNA]/[tmRNA]_TOT = [SmpB]/([SmpB] + Kd). Using this relationship, calculate the fraction of tmRNA bound for SmpB concentrations equal to 10⁴ K_d, 10³ K_d, 10² K_d, 10¹ K_d, K_d, 10^–1 K_d, 10^–2 K_d, 10^–3 K_d, and 10^–4 K_d.

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1.0 0.75 0.5 0.25 10-11 10-9 concentration of SmpB (M) 10-7 10-5 fraction bound