5.3 Effective concentration
5.3.1 Chain statistics and the J Factor
Consider the interaction between two complementary ends. The free energy change upon binding is [28]
∆G= ∆Ubond+ ∆Gconfig (5.27)
where ∆Ubond is the binding energy of the bond and ∆Gconfigis the change in chain-configurational free energy due to the formation of the bond. We consider the free energy change for both cyclization and dimerization reactions
∆GC ≡ −kTlogKC, (5.28)
∆GD ≡ −kTlogKD[L], (5.29)
±
Ω±V
Figure 5.11: A schematic picture of a DNA operator binding to a protein binding site. In the J factor calculation, we imagine a range of configurations in which DNA can bind. This range is given by a volumeδV and an orientational volumeδΩ. In the calculation, we assume is that this range is small. These parameters divide out of the resultingJ factor. In the rightmost panel, we display the intuitive picture of effective concentration. The DNA is fixed at one end, resulting in an effective concentration of the operator (blue dot).
where we have introduced a factor of the monomer concentration into the dimerization free energy definition so that the derivations for the two free energies are analogous.
We shall denote the generalized coordinates describing the relative displacement and orientation of the end{qi}. In our case, these degrees of freedom include the relative spatial displacement of the ends, the tangent of the polymer, as well as the relative twist [26].
The probability of the chain having some particular end configuration, specified by the coordi- nates{Qi} and within admissible range{δqi}, is
P =ρ(Qi)Y
dqi, (5.30)
where ρ is the probability density with respect to the coordinates {qi}. The free energy change associated with assuming this configuration is therefore
∆Gconfig=−kTlogρ(Qi)Y
i
δqi. (5.31)
Note that this free energy depends on the admissible range {δqi} which is not directly observable [26]. This scenario is drawn schematically for a DNA-protein complex in Fig.5.3.1.
For DNA hybridization, the ends of the DNA must not only be spatially coincident, but their tangents must be aligned, and the twist of the helix must also be in registry. Therefore theqi and
Qi are:
qi = (~x,cosθ, φ, ψ) (5.32)
Qi = (0,1,0,0) (5.33)
where ~xis the end-to-end displacement, θ andφ describe the orientation of the final tangent with respect to the initial tangent, andψis the twist mismatch. TheQiare the values of these coordinates required for hybridization.
We can can now write theJ factor in terms of the free energy change Eq.5.27:
J = [L] exp[−(∆GC−∆GD)/kT]. (5.34)
We shall assume that the hybridization energy is identical in the two configurations. TheJ factor then depends only on the chain-configuration free energy
J =ρC[L]/ρD, (5.35)
where ρC and ρD are the probability densities for cyclization and dimerization respectively. ρC
depends on the chain statistics but dimerization probability density is simply the monomer concen- tration times the orientational density which is isotropic in solutionρD= [L]/(4π·2π). TheJ factor is therefore [26,27,29,18]
J= 8π2ρC, (5.36)
whereρCis simply the polymer distribution function evaluated for the cyclization end configuration.
For the most part, we shall be interested in the mechanics of DNA when the twist is unobservable.
Of course, the twist is observable inJ factor, since DNA can only hybridize in twist registry, much like an electric cord will not plug into the wall unless it is in twist registry with the wall socket. We can approximately integrate out the twist degree of freedom by averaging theJ factor over a helical repeat since the helical repeat is 10 bp and reasonably small compared with the total length of the sequence. In this case, theJ factor becomes [29,18]
J = 4πρ0C, (5.37)
where ρ0C is the probability density summed over the twist degrees of freedom, which results in a twist free theory.
To generalize this calculation to protein-induced DNA looping, we need only change the definition of ρC. That is, if the protein is stiff compared to DNA, it determines configuration of the DNA operators. Instead of the probability density for the cyclization boundary conditions, the protein
Figure 5.12: The results of a cyclization assay for DNA sequences of 116 bp and 94 bp. Gel electrophoresis is exploited to separate the bands of different length and topology. Above,Lis linear monomer,Cis cyclized monomer,LDis linear dimer,LT is linear trimer,CDis cyclized dimer, etc.
The DNA is radio labeled and the concentration of DNA in each band is determined by the band intensity. Note that in the Ligase−column (Ligase is absent), there is only linear monomer. The unligated sequences are not stable enough to maintain their structure when run on the gel. For the 116 bp sequence, there are three Ligase + columns run at different DNA concentrations. As the DNA concentration is reduced, the bands corresponding to cyclized monomer increase in relative intensity while the bands corresponding to linear dimer decrease in relative intensity in agreement with the predictions of the kinetic equations. This concentration dependence is just one consistency check that the bands are correctly labeled. Note also that the bands corresponding to linear sequences run at the correct speed relative to the base pair ladder on the right-hand-side of the gel. This gel is from Ref. [2].
would dictate some more general set of boundary conditions which would give rise to an effective concentration. The relation between this loopingJ factor and the equilibrium constant is analogous to that derived for cyclization. For instance for the lac repressor looping reaction shown in Fig.5.1.2 panel B, the looping equilibrium constant is
Kloop= [looped]
[unlooped] =KopJ0, (5.38)
where J0 is the J factor evaluated for the correct operator-binding end configuration and Kop is the equilibrium constant for the repressor binding the free (unlooped) auxiliary operator sequence.
The looping equilibrium constant depends on both DNA mechanics (J0) as well as chemistry (Kop) [13,16,11,30].