Question

The biological roles of complex organic molecules are determined by their shape -- the way atoms and electrons create charge distributions on the molecule's surface. As a result of those charge distributions, the molecules can interact in specific ways, attracting and sticking to each other or to other organic molecules. For some proteins, their shape can be changed by the addition of a small amount of energy. Since different shapes can result in different biological roles, the same molecule can play different biological roles if the energy needed to change their shape is not too large.

Since the probability of a molecule having a higher energy than its ground state in a thermal bath is proportional to the Boltzmann factor, e=ΔEkBT, we will use this to generate estimates of the probability that a protein has a changed shape.
Let's call the normal (ground state) shape of a molecule α (alpha). Suppose that adding an energy ΔE to a molecule will change it to a new shape, β (beta). For the purposes of this problem we'll assume that this is a two-state system. The molecule has to be in either state α or the state β. If the molecule is immersed in a thermal bath of temperature T, find an expression for the probability, Pα that the molecule will be in the state α and the probability Pβ that the molecule will be in the state β. Since there are only two states, your two probabilities should add to 1 (100%). Express each probability as a function of the symbol z = e-ΔEkBT.
What value of ΔE would a molecule in the thermal bath from part (c) need to have to be in shape β one quarter of the time? Give your answer in meV.

Answer 1

The probability of a protein molecule being in a particular state can be calculated using the Boltzmann factor. The probabilities of the two states α and β sum to 1, and z = e-ΔE/kBT is used to determine these probabilities. By setting the probability of state β to 1/4, we can calculate the needed energy difference ΔE in meV.

Step-by-step explanation:

The state of a protein molecule in a thermal environment can be described probabilistically using the Boltzmann distribution. For a two-state system where the protein can exist in either state α (alpha) or state β (beta), the probabilities of each state can be determined based on the energy difference ΔE and the temperature of the environment, T. The Boltzmann factor is used to express these probabilities, where z = e-ΔE/kBT is the factor that represents the likelihood of the protein being in state β.

The convenience of using z is that it simplifies the expressions for probabilities, which are Pα = 1/(1+z) and Pβ = z/(1+z). Since these probabilities must sum to 1, we can set Pβ = 1/4 to solve for ΔE that would result in the protein being in shape β one quarter of the time. Rearranging the equation and solving for ΔE gives us the value in milli-electron volts (meV).