Chemoreceptors on cell surfaces when bacteria sense nutrients in their environment they begin adding sensory proteins (chemoreceptors) from inside the cell into the cell membrane. there are a total of n free chemoreceptors inside a cell. the energetic cost of adding a single chemoreceptor into the cell membrane is e > 0. once they are inside the membrane, chemoreceptors have a pairwise interaction energy of v< 0. (a) write down the total energy for the configuration having n chemoreceptors in the membrane (where, of course, n < n). (b) using the energy in part (a), write down the canonical partition function, z (hint: your partition function should be a sum over n) (c) let the fraction of chemoreceptors in the membrane be φ n/n. in the thermodynamic limit (n, n » 1), find @f (the free energy per chemoreceptor for the configuration with fraction φ ! e where z dp @f(n v, e φ β) (d) plot β.f versus φ and explain your result physically. that is, explain where and why exhibits minima