Energy Landscape Theory, Funnels, Specificity, and Optimal Criterion of Biomolecular Binding

We study the nature of biomolecular binding. We found that in general there exists several thermodynamic phases: a native binding phase, a non-native phase, and a glass or local trapping phase. The quantitative optimal criterion for the binding specificity is found to be the maximization of the ratio of the binding transition temperature versus the trapping transition temperature, or equivalently the ratio of the energy gap of binding between the native state and the average non-native states versus the dispersion or variance of the non-native states. This leads to a funneled binding energy landscape.

Figure 1

Funneled energy landscape of the biomolecular binding: panel (a) shows the energy landscape of binding; panel (b) shows the density of states of the corresponding landscape; panel (c) shows the free energy profile for the corresponding landscape with respect to $Q$, the fraction of native contacts with a barrier between the native and the non-native states; panel (d) shows the free energy profile in $Q$ without a barrier. $\delta E$ is the gap and $\Delta E$ is the spread or the roughness of the energy landscape. $Q_g$ is where the local trapping (local glass transition) occurs.

Figure 2

Phase diagram of biomolecular binding: the vertical axis is the temperature scaled gap or the slope, and the horizontal axis is the temperature scaled roughness of the energy landscape.

Figure 3

Energy spectra and chemical structures for the pipecolinyl fragment (left) and the second fragment (right) of size 14 heavy atoms, representative of molecules randomly taken from the commercially available Fine Chemicals Directory (MDL Information Systems, Inc., San Leandro, CA). Energies of structures in the native binding mode, where $R_{\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}=1.5\mathrm{\AA }$ are collapsed into a single value. The stability gap $\delta E$ between the binding mode and the average binding mode for the pipecolinyl fragment is indicated.