1. Questions in this problem concern protein molecules embedded within the membrane of a spherical vesicle, as sketched below.
The radius R of the sphere is fixed unless stated otherwise. Each protein molecule can move about freely on the surface of this sphere. The number of proteins per unit area is sufficiently low that interactions between different molecules can be neglected. Quantum mechanical effects can also be neglected throughout this exam, though we will " count" classical microstates according to dlta x*dlta v=h/m, where dlta x is resolution for enumerating positions, dlat v is the resolution for enumerating velocities, m is the mass of a protein molecule, and h is Planck's constant. This system can exchange energy with its environment, which serves as a heat bath at temperature T.
To start, consider a small patch a of the vesicle's spherical surface.
The location and size of this patch are described by values polar angle theta (which ranges from 0 to Pi ) and the azimuthal angle phy ( which ranges from 0 to 2 theta ). Specifically, a includes all points on the surface with polar angle between theta and theta + dlta theta and with azimuthal angle between phy and phy + dlta phy. We have in mind that dlta theta and dlta phy are both very small,m so that the patch a is essentially flat and rectangular. In this case the area of a is R^2 sin theta dlta theta dlta phy.
(a) Focus first on a single protein molecule confined to the patch a. Calculate the canonical partition function qa for this molecule's motion within the patch. Keep in mind that we are treating a as flat rectangle. Include contributions from fluctuations in its positions as well as its velocity.
Your answer should involve some, but not necessarily all, of the following quantities : theta, phy, dlta theta, dlta phy, the vesicle radius R, temperature T ( or inverse temperature beta ), the protein mass m, the thermal wavelength = hsquire root(2pi*KBT), fundamental constants.
(b) Now image the ring formed by a collection of patches a1, a2, a3, ...., an with same value of theta (and therefore the same area) and azimuthal angles phy = dlta phy , 2 dlta phy, 3 dlta phy, .... , n dlta phy, ..., n dlta phy, where n=2 pi dlta phy.
Calculate the partition function q(theta) for a single protein molecule confined to move within this ring at polar angle theta .
(c) Using your result from part (b), calculate the partition function q for a single protein molecule that can move freely over the entire sphere. by summing up contributions from rings at different values of theta. Take dlta theta to be sufficiently small that a sum over dlta can be replaced by an integral:
dlta (theta)f(theta)=integration from 0to pi (theta) f(theta), for any function f(theta).
(d) Using your result from part(c), calculate the canonical partition function Q for the collection of N identical,non-interacting proteins moving on the vesicle's surface.
(e) By differentiating Q from part (d) appropriately, calculate the average energy (E) and heat capacity Cv for a collection of N identical, non-interacting protein moving on the vesicle's surface.
(f) Next we will add contributions from the potential energy U due to the surface tension of the membrane,
U=yR^2
Here,y is a positive constant so that U increases with expansion of the vesicle. This potential energy has the same value for all protein microstates with the same vesicle radius R. For a given value of the vesicle radius R, calculate a partition function Q (R) for a collection of N membrane-embedded protein molecules. Your answer should account for contributions from the proteins' motion, as well as from the potential energy U. Keep in mind that U does not depend on the coordinates or velocities of any particle.
(g) Now imagine that the vesicle radius is allowed to fluctuate. Using your result from part (f), calculate the probability P (R1) of observing a radius value R1, relative to the probability P (R2) of observing a different radius value R2. You should be able to express P (R1)P(R2P) solely in terms of R1,R2,N,Y, and inverse temperature beta.(h) The probability P(R) has a local maximum at some radius value R*. Determine the value of R*! Your result should indicate that R* increases with temperature T and the number proteins N. Explain this result in terms of energy, multiplicity, and Blotzmann distribution
2. Building on your results from problem 1, we will now imagine that the vesicle lies near as substrate, to which the embedded proteins are attracted, as sketched below.(for simplicity only one protein molecule is shown.) We will assume the vesicle itself to be fixed, but proteins molecules can change their distance from the substrate by moving on the vesicle's surface.In other words, as the angle theta and phy defining a protein's position change, so may its distance d from substrate. Specifically, d=d0-R*cos(theta), where d0 is a constant independent of theta and phy. The attraction u between a protein and substrate is strongest at closet approach(theta=0) and becomes weaker with increasing distance (as theta increases). We will take a very simple form for this dependence(plotted below), u(theta)=-e cos(theta), where e is a positive constant. This attraction u(theta) is the only source of potential energy,i.e., do not worry about the vesicle's surface tension we considered in previous problems. Take the vesicle radius R to be fixed throughout this: (a) Describe your expectations for the distribution of proteins on the surface of the vesicle, as temperature is increased from a low temperature T1<>e/KB. What do you expect for the average energy of interaction of a single protein with the substrate, as temperature is increased from T2 to T1?
(b) calculate the partition function qa for a single protein that is confined to a very small patch a on the vesicles surface. As in problem 1, a includes all points on the surface with polar angle between theta and theta+ dlta theta and with azimuthal angle between phy and phy+ dlta phy. Consider dlta theta to be sufficiently small that u(theta) does not vary within the patch.
(c) summing over possible values of theta and phy, calculate the partition function q for a single protein that moves over the entire vesicle surface while biased by the attraction u(theta). Again, take dlta theta to be sufficiently small that a sum over theta can be safely replaced by integral.
(d) By differentiating q appropriately, calculate the average energy for this single protein.
(e)The average total energy of a single protein, as calculated in part(d),contains contributions from the kinetic energy of the protein and its interaction u with the substrate. Using a computer, plot the average total energy and average potential energy as a function of temperature, in the same plot. Use units of e/KB for temperature, and units of e for the energy.
(f) Using your result from part(e), sketch(no need to use a computer!) the heat capacity C (in units of KB), as a function of temperature (in units of e/KB), for a single proteins moving on the vesicle surface and interacting with the substrate. Explain the values of C in the limit of very low and high temperatures!