Phys 139: Problem set 5
Due by 11:59pm on Monday, May 13
May 7, 2024
1. PBoC 15.5
2. Consider a single-molecule experiment in which kinesin is hindered by an applied force f by attaching the motor to a plastic bead, and manip- ulating the bead using an optical trap. Recall that, in the Bell model, a chemical rate involving motion by a distance δ in the opposite direction of applied force is slowed according to k(f) = k(f = 0) exp(-f δ/T); inversely, a rate involving motion in the same direction as the applied force is sped up by k(f) = k(f = 0) exp(+f δ/T).
Analyze the efect of force on the Michaelis-Menten parameters, v max and KM, in the kinesin velocity versus [ATP] curve, in the presence of a hindering load force f. Particularly, determine if each parameter increases or decreases with force if we assume (1) Kinesin steps forward a distanced = 8 nm upon ATP binding, with no motion upon catalysis; or (2) Kinesin steps forward by d = 8 nm upon catalysis, with no motion upon ATP binding. Compare your prediction to experiment by analyzing the data used in Fig. 16.34(a) in the text (an excel spreadsheet with this data has been uploaded in the modules). Does either model work? What does this mean for motor activity?
3. In class, we worked out the probability distribution of step numbers, N , for a motor protein after a fixed time, τ , i.e. P (Njτ ). Here, the goal is to work out, and apply, the probability distribution of times to arrive at a fixed step number, P (τ jN).
(a) Assume that each step involves a single rate-determining reac- tion, with rate k. To take 2 steps in time τ is proportional to the probability of taking 1 step in time t, , multiplied by the proba- bility to take a second step in time τ - t, , and integrated over all choices oft, (this sort of integral is called a convolution). Use this to find P (τ j2).
(b) Extend the argument from the previous part to find P (τjN). This can be done by iteratively working out the distributions for N = 3, 4, . . . and detecting the pattern. Alternatively, it hap- pens to be the case that convolutions can be quickly carried out through multiplication of Fourier (or Laplace) transforms; if you feel confident in your transform mathematics, you could try this approach.
(c) Calculate the time-domain randomness involved with taking N steps, assuming that each step involves either 1 reaction per step, or 2 sub-reactions (of equal rate k). Note that the time domain randomness is defined as
where τ1 is the mean time to take one step.
4. PBoC 16.2. Best to do this after the previous problem. Note that Myosin V walks along actin in a manner similar to kinesin walking along a microtubule, and analogous to human walking (i.e. it has two ‘feet’ each connected to ‘legs’, and it walks along actin with an alter- nating left-right stepping pattern), and that the ‘light chain domain’ is the proper name for the part of the protein corresponding to the ‘leg’ .