Are DNA Transcription Factor Proteins Maxwellian Demons? * physics as a design constraint on molecular evolution. Alexander Grosberg, Longhua Hu, U. Minnesota, NYU U. Minnesota Biophysical Journal 2008 Aug;95(3):1151-6.
Escherichia Coli lactose glucose Enzymatic digestion of lactose -> glucose: genetic control * Switching time: minutes
E. coli: very crowded interior Ribosomes gene transcription: RNA Polymerase mrna DNA 10 7 base-pairs
Protein Synthesis * only if lactose concentration is high: need a switch.
Genetic Switch Transcription Factor: lacr RNA Polymerase Promoter Site Transcription Factor Genetic code of enzyme Binding Site: Operator Sequence 15-30 bases
lacr Repressor: specific binding lactose binding 4 Reading Heads operator sequences
Direct Read-out amino-acid: Arginine nucleic acid: Guanine
Reading Rate > DNA 10 4 Searching bases per second per lacr. lacr is passive : Diffusion * about 10 7 possible binding sites. operator sequence * about 10 2 repressor proteins. * Reaction rate = on-rate k a times concentration lacr. k a 10 10 M 1 sec 1
Diffusion-Limited Reaction Rates Diffusion Equation: c t = D 3 2 c ξ Reaction Radius : reaction rate slowest step. The diffusion-limited reaction rate equals the steady-state diffusion current J from infinity into an absorber with radius ξ. (Pontryagin et al., 1937)
Smoluchowski speed limit: Kinetic Perfection R. von Smoluchowski (1915) Reaction Rate k a : on-rate J = 4π D 3 ξc 3D diffusion coefficient (D 3 3 10 8 cm 2 / sec) reactant concentration: (100/µ target radius? 3 ; 10-7 M) DNA radius/protein size b ~ nm maximum on-rate: In-vitro on-rate: k a 10 9 M 1 sec 1? k a 10 10 M 1 sec 1
Slide-Skip Diffusion (Berg, Winter & von Hippel Biochem., 1981) lacr Weak non-specific affinity: Combined 1D - 3D diffusion. Searched DNA section L(Time) Time (not Time 1/2 ) operator sequence * D 1 0.4 10 9 cm 2 / sec 10 2 D 3 (in-vivo, Elf et al., Science, 2007)
Non-specific electrostatic affinity positive charge negative charge operator sequences
maintain Smoluchowski relation T. Hu, A. Y. Grosberg, and B. I. Shklovskii, Biophys. J. 90, 2731 (2006). c R ( ) Equilibrium concentration of free proteins. Law of Mass Action Adsorber Sphere : non-equilibrium, D=1 diffusion J = 4π D 3 ξc R ( )
Law of Mass Action lacr + DNA lacr DNA c R ( ) c R c R DNA ( )φ ( ) = K non-specific equilibrium constant ( 10-3 ) K exp( ΔG / k B T ) c R DNA c R ( ) : Concentration of DNA-bound proteins. φ : DNA volume fraction of E.Coli ( 10-2 ). ( ) : Concentration of free proteins. c R ( ) = c 1+ φ / K c = c R ( ) + c ( R DNA )
c R DNA ( ) ( ) = φc R K Outside target sphere: bound lacr line density ρ c R DNA b 2 Inside target sphere: 1D concentration gradient ρ ξ perfect sink ξ DNA radius 1D diffusion current to target: J D 1 ρ ξ b Equate 3D & 1D currents: J D 3 ξc R ( ) 2 c D R DNA 1 ξ ( )
D ξ b K 1 1 D 3 New Speed-limit Antenna Length naive limit Enhancement Factor Optimal Design K = φ J / ( D 3 bc) D 1 D 3 K K + φ 10 3 10 2 D 1 /D 3 ~1/50 why? φ K
LacR-Operator lacr binding: Repressor: large conformational dimer-dimer change: 10-3 sec Reading Heads operator sequences
We assumed perfect DNA scanning by lacr! D 1 0.4 10 9 cm 2 / sec (measured diffusion constant) D 1 b2 τ base-pair spacing: 10-10 m 10-6 sec. Large D 1 : overshoot. Reduces read-out fidelity
How does lacr/dna read-out actually work? lacr non-operator DNA Coulomb interaction. no direct contacts between amino-acids and nuclear bases. indirect read-out : Sequencedependent bending rigidity. Thermal fluctuations (Kalodimos et al., Science, 2004)
Read-out by conformational switching hinges Reading heads + *mobile *immobile. *indirect read-out. *partially folded. 0 *immobile. *direct read-out. *fully folded.
Proposal (Mirny et al.): does lacr achieve high search efficiency by a clever choice for these conformational fluctuations? Employ thermal fluctuations as a DNA scanner?
( ) D 1 D 1eff = p + Ω eff = p( )Ω Two-Channel Model high transport rate : high read-out fidelity : p ( + ) 1 p( ) 1 D 1 ~D 3 + + + + + + + + + Boltzmann Factor: 0 Ω ΔE ± Reaction Rate: non-specific bound state to specific bound state (slow: khz ) p ( + ) / p( ) = exp( βδe ) ± = µ
J ( ω ) / ( D 3 bc) D 1 D 3 K K + φ 1 ω ω µ 2 1+ µ + µ 2 µ Dimensionless Transition Rate fidelity factor ω = 2Ωb 2 ( K D 1 D ) 1/2 3 µ = exp ΔE ± k B T J ( ω ) / ( D 3 bc) J ( ω 0) / ( D 3 bc) 0 D 1eff D 3 K K + φ
µ opt = ( 1 + 2ω 1) / 2 ω =
How does lacr stack up? K ~ 10-3 ω 10-4 Ω (Hz) lacr approaches speed limit! exp ΔE ± k B T opt ( ) / 2 --> ΔE ± k B T = 1+ 2ω 1 lacr utilizes thermal fluctuations impedance matching optimal choice K and µ: reaction rate and diffusion rate