2757 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C7: BIOLOGICAL PHYSICS TRINITY TERM 2013 Monday, 17 June, 2.30 pm 5.45 pm 15 minutes reading time Answer four questions. Start the answer to each question in a fresh book. A list of physical constants and conversion factors accompanies this paper. The numbers in the margin indicate the weight that the Examiners anticipate assigning to each part of the question. Do NOT turn over until told that you may do so. 1
1. (i) Explain what is meant by an action potential and describe qualitatively the role of ion channels in the propagation of electrical signals in an unmyelinated axon. [6] (ii) Using classical cable theory show that for passive conduction the membrane potential V (x, t) is given by 1 ρ m (r m r a ) 2 V (x, t) V (x, t) 2 ρ a x 2 = V (x, t) + ρ m ε 0 ε m, t where ρ a and ρ m are the resistivities of the axoplasm and membrane respectively, r a is the axon radius and r m is the membrane thickness, ε m is the dielectric permittivity of the membrane and ε 0 is the permittivity of free space. Estimate the temporal and spatial decay lengths of an electrical impulse in a squid axon with r a = 500 µm, ρ a = 0.6 Ω m, r m = 5 nm, ρ m = 1.5 10 8 Ω m, ε m = 5. [10] (iii) An action potential travels along the axon with a fixed shape at a velocity v, such that V (x, t) = Ṽ (t x/v), where Ṽ (t) = V (0, t). The time course Ṽ (t) is shown schematically in the diagram: Using the relation given in (ii), obtain an expression for the membrane current and reconstruct its qualitative profile I(t) from the time course Ṽ (t) shown in the diagram. Indicate on your diagram where the different ion channels contribute to the observed membrane current. [9] 2757 2
2. Arginine (Arg) is a hydrophilic amino acid that remains positively charged in most acidic, neutral and basic environments. Estimate the energy required to transfer it from water to a lipid bilayer. [Arginine molar volume = 1.26 10 4 m 3.] [4] Describe the mechanism which voltage-gated ion channels use to respond to changes in transmembrane potential. Your answer should include examples of experimental validation of the structural mechanisms proposed for voltage-dependent gating. [12] The open and closed states of a voltage-gated potassium channel are separated by a conformational energy barrier W such that the ratio of open to closed states can be expressed as exp( W/k B T ) when the membrane potential is zero. The transition from the open state moves a gating charge z g across the membrane. Obtain an expression for the fraction of open channels when a voltage V is applied. [5] The figure below shows the relative displacement current activated by a brief voltage pulse. Estimate the gating charge in the limit W z g V, and comment on your result. [4] 2757 3 [Turn over]
3. Describe a single molecule force spectroscopy experiment where an AFM tip is used to unfold a single molecule of the membrane protein bacteriorhodopsin, which is depicted in the figure below. Include a simple schematic of the main components of the experimental set-up, describe how the AFM tip is controlled and how it is used to plot the characteristic force vs. distance curve as the protein unfolds. [10] The figure below shows a typical force vs. distance curve corresponding to the unfolding of single bacteriorhodopsin proteins at two salt concentrations, 20 and 40 mm KCl, respectively. Describe and interpret qualitatively the principal features of both curves and discuss the effect of the ionic concentration. [Hint: Consider interactions between the protein and its environment within the hydrophobic core of the membrane and when unfolded in the surrounding solution.] [9] The Worm Like Chain model describes a protein as an isotropic, homogeneous elastic rod whose trajectory varies continuously and smoothly through space. This model gives the following formula relating force (f) to extension (r) (r is equivalent to distance in the figure above): f(r) = k BT p r L contour 1 ( ) 2 4 1 r L contour 1, 4 where p = persistence length, the length over which statistical segments remain directionally correlated in space and L contour is the contour length, i.e. the polymer length at maximum physically possible extension What physical information can be extracted from fitting each peak of the experimental curves with this model? [6] 2757 4
4. Draw a schematic describing the basic organization of nucleic acids and proteins within a bacterial transcription complex. Discuss the role of sigma factors in transcription. What is the sequence of the first hexanucleotide RNA synthesized from a gene starting with the sequence 5 AGTCATTCG 3 on the non-template strand? [8] RNAP searches for its target site by combining 1D and 3D diffusion with respective diffusion coefficients D 1 and D 3, resulting in a mean search time of τ 1+3 = Lλ + V D 1 D 3 λ, where V is the cell volume, L is the chromosome contour length, and λ is the characteristic 1D sliding length due to 1D RNAP diffusion on DNA before dissociation. Sketch and discuss the dependence of τ 1+3 upon λ, and explain the origin and dependence of the two terms contributing to τ 1+3 as well as their functional significance for the combined search. Find an expression for the λ associated with the minimum τ 1+3. [6] The stability of the double-stranded DNA region around the transcription start site modulates transcription initiation. Consider a zipper model (akin to the one describing helix-coil transitions in polypeptides) for the formation of base pairs between a 5 AGTCATTCG 3 DNA and its complementary strand. If the base-pairing initiation and propagation steps are characterized by equilibrium constants κ and s, respectively, show that, for complementary single strands of DNA with N nucleotides each, the partition function q for the ensemble of partially and fully double-stranded DNA structures with j base pairs (considering only states with j 1) is q = κ and that the series can be reduced to N (N j + 1) s j 1, j=1 q = κ [ s N+1 (N + 1)s + N ] /(s 1) 2. Write partition function expressions for s 1 and discuss the result with regard to the types of DNA surrounding the start site in this limit. [11] 2757 5 [Turn over]
5. Make a labelled sketch of the enzyme ATP-synthase, and describe briefly its mechanism and its function in living cells. [8] In a single-molecule experiment, the γ-subunit of F 1 -ATPase, isolated from ATPsynthase, is labelled with markers of negligible size. When ATP is added at an appropriate concentration, markers rotate in alternating large and small steps. Time intervals τ before the smaller steps are distributed as P (τ) = k ak b (k a k b ) [ exp{ k bτ} exp{ k a τ} ], where k a and k b are constants. What can be deduced from this observation about these steps? Include a derivation and a sketch of the distribution P (τ) in your answer. [9] At a fixed ATP concentration, the average time interval before the larger steps in the experiment above is 10 ms. When a constant external torque of 10 pn nm is applied to the gold particle opposing its rotation, the average time interval increases to 40 ms. Draw a labelled diagram of free-energy versus angle for the process that causes the larger step, showing free-energies both with and without the external torque. Hence, or otherwise, estimate the angle rotated by the gold bead to reach the transition state from the state before the step. [8] 2757 6
6. Biological cells store free energy in the form of sugars (e.g. glucose), high energy molecules such as NADH and ATP, and electrochemical gradients. Sketch and discuss an overview of the ways in which these forms of free energy are interconverted and utilized, including in your sketch the primary source of biological free energy. [10] A future nanotechnologist designs a device to generate an electrochemical potential (ion-motive force, IMF ) of K + ions across the membrane of a phospholipid vesicle, using a reversible ATP-driven potassium pump that couples hydrolysis of 1 ATP molecule to pumping of m ions (see the figure below). Initially, the pump is switched off, the number of [K + ] ions inside the vesicle is N, internal and external [K + ] are equal, the membrane voltage ψ = 0 and there are no ADP nor phosphate ions inside the vesicle. The pump is started and n potassium ions are pumped into the vesicle. Derive expressions for the free energy of ATP hydrolysis inside the vesicle and the IMF in terms of N, m, n, the initial number X of ATP molecules in the vesicle, the membrane capacitance C, the vesicle volume V, the standard free energy G 0 of ATP hydrolysis per molecule, the temperature T and any necessary universal constants. [9] A similar device is generated where the pump is replaced by potassium and sodium channels. These generate a relative permeability across the membrane of P K /P Na = 50. The inside of the vesicle contains 120 mm KCl and 12 mm NaCl and the vesicles are then placed into a solution containing 140 mm NaCl and 5 mm KCl. Calculate what happens to the potential difference across the membrane and describe the significance of your findings. [6] 2757 7 [Turn over]
7. For the gene regulatory motifs shown in the figures below, transcription factor X activates the gene for protein Y, and both X and Y activate the gene for protein(s) Z k with OR gate logic. The ligand S Y that changes Y to its active state is always present. The production rates β i are equal to each other as are the degradation rates α i. You can use the logic approximation throughout this problem. (a) For motif (a), X is always present at its steady state concentration; all activation thresholds are equal: K ij = 0.5β i /α i. Calculate the time at which Z reaches half its steady state concentration following an ON step where S x appears in the system, and after an OFF step where S x disappears. Sketch the dynamics of the protein concentrations. Assume that that Z 1 reaches steady state between the ON and OFF steps. Give a biological reason for evolving this kind of network motif in bacteria. [5] (b) For motif (b), S x is always present. Calculate thresholds K ij such that the genes are turned on in the order Z 1, then Z 2 and then Z 3 in response to an ON step where X production is first activated by another promoter, and subsequently turned off in the order Z 1, then Z 2 and then Z 3 in response to an OFF step where X production stops. Sketch the protein concentrations. Give a biological reason for evolving this type of network motif in bacteria. [5] (c) In motif (c), Y represses X if Y s concentration is above a threshold K Y X ; all thresholds are equal: K ij = 0.5β i /α i ; X is initially present at its steady state concentration. Calculate how long after an ON step, where S x first appears, Z 1 takes to reach its first concentration maximum Z1 max1. Calculate Z1 max1. Calculate the same properties if the motif is changed so that X and Y activate Z 1 with AND logic. [8] (d) The maximum adsorption rate of a sphere of radius R, in an infinite medium containing particles with diffusion coefficient D and at an initial concentration C 0, is I 0 = 4πDRC 0. The maximum adsorption rate of a disc of radius S in the same medium is I D = 4DSC 0. Consider a spherical bacterium of radius R = 1 µm that has N disc shaped receptors of radius S = 1 nm on its surface. The total adsorption rate of the N receptors can be approximated as: [ ( )] I 1 I0 = 1 + f. I 0 NI D Make a simple physically motivated approximation for the function f(x) and hence solve for the number of receptors for which I = 1 2 I 0. What fraction of the surface area of the bacterium is then covered by the receptors? Discuss the biological relevance of this result for bacteria. [Hint: to find f(x) consider the limits of large and small N.] [7] 2757 8
8. Write short notes on two of the following: (a) Outline the RNA world hypothesis and its justification. (b) Describe the use of single molecule fluorescence to monitor real-time synthesis of individual protein molecules in living bacteria. Illustrate your answer with appropriate schematic diagrams. (c) Describe the structural and biophysical principles which ion channels use to distinguish between different ions. [25] 2757 9 [LAST PAGE]