Villa et al. (2005) Structural dynamics of the lac repressor-dna complex revealed by a multiscale simulation. PNAS 102: 6783-6788. Background: The lac operon is a cluster of genes in the E. coli genome that encode proteins involved in lactose metabolism. The lac repressor (LacI) is a DNA-binding protein that inhibits the expression of the operon in the absence of lactose. This study used a combined atom-filament model of a LacI-DNA complex to investigate the structural dynamics of the repressor. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 1 / 24
The lac operon contains three protein-coding genes: lacz encodes β-galactosidase, which cleaves lactose into galactose and glucose; lacy encodes a lactose permease, which pumps lactose into the cell; laca encodes a thiogalactoside transacetylase. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 2 / 24
The lac genes are co-regulated. The three lac genes share a common promoter and are transcribed as a single polycistronic mrna. The lac operon is transcribed at very low levels whenever lactose is absent. (This is thought to prevent the wasteful production of enzymes that aren t needed.) The operon is also transcribed at low levels when glucose is present, even if lactose is also present. (Glucose may be a preferred source of energy.) The lac operon is transcribed at high levels if lactose is present and glucose is absent. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 3 / 24
LacI inhibits transcription when lactose is absent. In the absence of lactose, LacI binds to two operator sequences that flank the promoter. This interferes with binding of RNA polymerase to the lac promoter and blocks most transcription of the operon. Figure from Wilson et al. (2007) Cell. Mol. Life Sci. 64: 3-16. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 4 / 24
The operon is weakly transcribed when glucose is present. Allolactose is a product of lactose metabolism, which occurs at low levels even when the repressor is bound to the operator. Binding of allolactose to LacI lowers its affinity for the operator. However, dissociation of LacI from the operator only results in a small increase in lac transcription if glucose is present in the cell. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 5 / 24
The operon is strongly expressed if lactose is present and glucose is absent. Absence of glucose within the cell leads to an increase in cyclic AMP levels. Binding of camp to the catabolite activator protein (CAP) increases the affinity of this protein for a region upstream of the lac promoter. CAP binding greatly facilitates transcription of the lac genes. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 6 / 24
LacI forms a tetramer that binds to two operator sequences. Each monomer contains a HTH motif that binds to the major groove of the DNA. The lac operon has three operator sequences. The tetramer binds to two of these sites at a time, causing the intervening DNA to fold into a loop of 75 or 384 bp. LacI must be flexible enough to search for the operators but also strong enough to withstand the tension created by the DNA loop. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 7 / 24
The lac operon functions as a metabolic switch. Bistability occurs in the presence of lactose: the operon can be on or off. Switching is triggered by dissociation of LacI from both operator sequences. Fig. 4 from Choi et al. (2008), Science 322: 442-446. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 8 / 24
Multiscale model of a LacI-operator complex. The aim of this study was to investigate the effects of the DNA loop on the structural dynamics of the LacI-operator complex. However, MD simulations face two problems: The complete solvated LacI-operator system would contain between 700,000 and two million atoms, mainly contributed by the solvent. The loop dynamics occur on timescales of microseconds or longer, while all-atom MD models can be used to simulate nanosecond-scale processes. The authors approach is to use an all-atom model of the LacI bound to the recognition sites but model the intervening loop as an elastic ribbon. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 9 / 24
Molecular Dynamics Protocol The initial LacI-DNA complex was built from several crystal structures and embedded in a periodic box of water and salt. Two systems were simulated - one containing 226,314 atoms and the other containing 314,452 atoms. The difference is due to size of the water box and the number of salt molecules. Each system was simulated at constant temperature T = 300 K and pressure P = 1 atm. The smaller system was simulated for 22.4 ns, while the larger was simulated for 17 ns. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 10 / 24
Elastic Rod Model of DNA The 75 bp loop was modeled as an elastic rod connected to the terminal base pairs of the LacI binding sites. The equilibrium structure of the loop was determined by numerically solving the Kirchoff equations of elasticity. The Kirchhoff equations describe the kinematics of an elastic rod: each set of boundary conditions determines a single, equilibrium structure which balances the elastic forces caused by bending and twisting at every cross-section. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 11 / 24
Parametrization of an elastic rod model of a DNA helix An elastic rod can be represented by its centerline r(s) and a local coordinate frame {d 1 (s), d 2 (s), d 3 (s)}. s is arclength along the centerline. d 1 (s) and d 2 (s) point along the major and minor axes of the base pair at s. d 3 (s) = d 1 (s) d 2 (s) points in the direction of the centerline. The model employed here assumes that the rod is inextensible: ṙ = d ds r = d 3. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 12 / 24
Curvature and Twist The angular velocity of the local coordinate frame can be written as ḋ i = k d i where k = (K 1, K 2, Ω) is the vector of strains; K 1 (s) and K 2 (s) are the curvatures; Ω(s) is the local twist. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 13 / 24
At equilibrium, the elastic forces N and torques M balance the external forces f and torques g acting at each point of the centerline: Ṅ + ḟ = 0 Ṁ + ġ + ṙ N = 0 Here N = i N id i and M = i M id i, where N 1, N 2 are the shear forces; N 3 is the force of tension or compression; M 1, M 2 are the bending moments of the principal axes; M 3 is the twisting moment. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 14 / 24
The Bernoulli-Euler approximation assumes that the elastic torque depends linearly on the local curvatures and the twist: where M = A 1 κ 1 d 1 + A 2 κ 2 d 2 + Cωd 3 A 1, A 2 are the bending rigidities; C is the twisting rigidity; κ i (s) = K i (s) κ 0 i (s) is the deviation from the intrinsic curvature of the rod; ω(s) = Ω(s) ω 0 (s) is the deviation from the intrinsic twist of the rod. The intrinsic curvatures and twist are the values adopted by an energy-minimizing rod in the absence of any external forces. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 15 / 24
Kirchhoff s equations characterize the geometry of an elastic rod at equilibrium under the Bernoulli-Euler approximation. These can be written as a 13-dim system of nonlinear ODEs: see eqns (2.39) - (2.49) in Villa et al. (2004); Multiscale Model. Simul. 2: 527-553. The equations can either be derived directly from the local balance conditions on the force and torque, or from the Euler-Lagrange equations for the energy functional: U = 1 2 L 0 [ β1 (s)κ 2 1(s) + β 2 (s)κ 2 2(s) + γ(s)ω 2 (s) ] ds. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 16 / 24
Coupling of the MD and DNA Loop Model The loop model was coupled to the MD simulation by applying the forces generated by the elastic stress and torque of looping to the operator ends. The equilibrium loop structure was recomputed every 10 ps using B.C. s derived from the last MD structure. Thermal fluctuations of the DNA loop were modeled by subjecting the loop to a continuously varying Gaussian force of magnitude 80 pn. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 17 / 24
Geometry of the LacI-Operator Complex Several summary statistics were monitored during the simulations: the cleft angle α measures the opening between the two dimer arms; θ i is the altitudinal angle of head group i; φ i is the azimuthal angle of head group i. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 18 / 24
Fluctuations of the DNA loop are mainly absorbed by the head groups. Fig. 2b shows the RMSD of the LacI core domains (top), of the head groups (middle), and of the head groups after alignment of the core domains. Fig. 2c shows the fluctuations of α (violet), θ 1 (blue), φ 1 (green), θ 2 (red), and φ 2 (orange). The two head groups tend to rotate in different planes. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 19 / 24
Head group rotation allows the DNA loop to relax. Fig. 4b shows a series of 200 ps snapshots of the complex: initial structures in red; final structures in blue. The loop energy decreases from 20 k B T to 12 k B T. Separation of the LacI-DNA binding sites can occur without an opening of the cleft. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 20 / 24
Opening of the LacI-DNA complex by an external force. 500 pn forces were applied to the outer DNA ends for 16 ns, eventually resulting in separation of the two LacI dimers (IV ). Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 21 / 24
LacI disassociation occurs in four stages. Fig. 4C shows the change in the cleft (violet) and head group angles (green). Stage 1: both angles increase gradually; Stage 2: α is constant, while θ increases rapidly; Stage 3: α resumes increasing; Stage 4: the rate of opening increases. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 22 / 24
The LacI tetramer is stabilized by several intermolecular interactions. Salt bridges E235 1B R326 2A and E235 2B R326 1A form during Stage II and then rupture during Stage III. A charge-dipole interaction between R351 1A and the α-helix 247 2B 260 2B is disrupted during Stage III. Hydrophobic interactions between two α-helices, 222 1B 236 1B and 222 2B 236 2B disappear by Stage IV. Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 23 / 24
Summary The mobility of the LacI head groups allows the enzyme to withstand the substantial stresses generated by the DNA loop formed between the two operator binding sites. Several salt bridges and helix-helix interactions stabilize the V configuration of the LacI tetramer during repression of the lac operon. The dynamics of large protein-dna complexes can be studied using multiscale models that couple atomic-level MD to mechanical descriptions of DNA loops. However, the disparity of the timescales of these different processes is not clearly addressed in this paper: can one simply accelerate the motion of the loop as done here? Jay Taylor (ASU) APM 530 - Lecture 14 Fall 2010 24 / 24