Protein folding and translocation : single-molecule
investigations
Leeuwen, Rudolphus Gerardus Henricus van
Citation
Leeuwen, R. G. H. van. (2006, November 16). Protein folding and translocation :
single-molecule investigations. FOM Institute for Atomic and Molecular Physics
(AMOLF), Faculty of Mathematics and Natural Sciences, Leiden University.
Retrieved from https://hdl.handle.net/1887/4991
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A
The worm-like chain (wlc) model
To model the extension of dsdna and polypeptides in our experiments, we used the worm-like chain model. We will present two variants of this model: the inextensible and the extensible worm-like chain model.
For the experiments that are presented in this thesis, knowing the elastic properties of double-stranded dna (dsdna) is extremely important. In Chapters 2 and 3, dsdna is used as a molecular spacer between the (pre-)protein that is being studied and the optically trapped microsphere that is used to measure forces. In Chapter 4, the shortening of dsdna as it is being packaged by a bacteriophage is measured. In the measurements presented in these chapters, dna compliance obscures the phenomena of interest: protein translocation, polypeptide extension after unfolding and packaging speeds, respectively. Knowing the elastic properties of dsdna, one can remove effects stemming from dna compliance from the measurements and focus on the phenomenon of interest.
In solution, a relaxed dsdna molecule bends and curves locally as a result of thermal fluctuations. Even when a small force is exerted on both ends, these fluctuations will cause the end-to-end distance to be smaller than the dna contour length L. This bending results in an elasticity that is purely entropic of origin.
For low forces (F<5 pN), dna elasticity can best be described by the inextensible worm-like chain (wlc) model. In this model, the dna is treated as a flexible rod of length L that curves smoothly as a result of thermal fluctuations. The rod’s local direction decorrelates at a distance s along the curve according to exp(−sP), where the decay length, P, is the persistence length. In this thesis, we used the value of the persistence length that was found by Smith et al. [103] and that is used most often in literature: 53 nm. Using the inextensible wlc model, the force required to extend a dna strand of contour length L to end-to-end distance x can be calculated numerically. Moreover, a useful approximation was given by Bustamante et al. [7]:
FP kBT = 1 41 − x L −2 − 1 4+ x L. (A.1)
At forces higher than ~5 pN, the dna force–extension curve deviates from the inextensible wlc model. At forces between 5 and 50 pN, the dna elasticity is no longer merely entropic. The dna can be extended here to extensions beyond
A. The worm-like chain (wlc) model
extension
force
Figure A.1:Force–extension curve of a dsdna molecule with a contour length of 920 nm (black). The experimental configuration is shown. The gray curve indicates an extensible worm-like chain (wlc) model with a persistence length of 53 nm, a contour length of 920 nm and an elastic stretch modulus of 1200 pN.
its contour length, indicating the existence of an intrinsic elasticity of the fully extended molecule. The extension x of a dna molecule that is being held by a force F can here be approximated using the extensible worm-like chain model [104, 100]:
x L=1 − 1 2 kBT FP 1~2 +F S, (A.2)
where S is the elastic stretch modulus of dna. In this thesis, a value of 1200 pN is used for S. This value is used often in literature (see, e. g., Smith et al. [5]) and gives good agreement with experimental data.
In Figure A.1, an example of a dsdna force–extension curve is shown, measured using our optical-tweezers setup. For this measurement, a dna molecule with a contour length L of 920 nm was tethered between an optically trapped microsphere and a microsphere that was held by a micropipette as shown. For the connections of the dna to the microspheres, biotin/streptavidin and digoxigenin/anti-digoxigenin interactions were used. By moving the micropipette away from the optical trap, the dna extension was increased and the trapped microsphere was slowly pulled away from the trap center by the increased force. This graph also shows the force– extension curve that is predicted by the extensible wlc model, showing a clear overlap between theory and experiment.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 −20 0 20 40 60 80 100 120 extension (µm) force (pN)
Figure A.2:Overstretching transition in a dsdna molecule with contour length 800 nm.
tensible wlc model with a persistence length of 0.5–2 pN and the force–extension curve can be calculated by regarding the dna and the polypeptide as two springs in series (FDNA(xDNA) = Fpolypeptide(xpolypeptide) and xDNA+xpolypeptide=xtot). For, polypeptides, we used the inextensible wlc model rather than the extensible wlc model because it gave better agreement with the data.
At forces of 65 pN or more, the extended dna molecule undergoes a so-called overstretching transition [105, 106]. In this transition, the dna molecule can be extended up to 70% beyond its contour length. Figure A.2 shows the overstretching transition of a 800-nm dna strand, tethered between two microspheres. In the overstretching transition, the canonical B-form dna is deformed into so-called S-form dna. It is thought that during the overstretching transition from B-dna to S-dna the double helix structure is slowly unwound. The force–extension curve of this S-dna resembles that of single-stranded dna.
