Excluded Volume Effects in Gene Stretcing Pui-Man Lam Pysics Department, Soutern University Baton Rouge, Louisiana 7083 Abstract We investigate te effects excluded volume on te stretcing of a single DNA in solution. We find tat for small force F, te extension is not linear in F but proportion to F γ, wit γ(-ν)/ν, were ν is te well-known universal correlation lengt exponent. A freely joint cain model wit te segment lengt cosen to reproduce te small extension beavior gives excellent fit to te experimental data of λ-page DNA over te wole experimental range. We sow tat excluded volume effects are stronger in two dimensions and also derive results in two dimensions wic are different from te tree dimensional results. Tis suggests experiments to be performed in tese lower dimensions. I. INTRODUCTION Many forms of polymers or macromolecules suc as proteins, lipids, fatty acids and DNA, are vital for life. Te DNA is a very large molecule. For instance te DNA in one uman cell can encode approximately 00,000 genes. It is partitioned into 46 cromosomes, 3 from eac parent and is normally twisted and folded into te nucleus of a cell a few micrometers in widt. In solution te uman DNA consists of two strands of polymeried nucleotides, twisted into a rigt and elix. New tecnologies makes it possible to study single molecules of DNA. Suc studies can give insigts for
development of laboratory tecniques for analying, fractionating and sequencing DNA. One suc study investigates ow a single DNA stretces wen it is pulled wit a force at its two ends, using micromanipulation tools suc as optical tweeers or microneedles to grasp te molecule by its two ends. Stretcing te DNA can provide a muc clearer picture under te microscope. As te average end-to-end separation increases, a force arises tat opposes te stretcing. Tis force is entropic in nature since te number of configurations of te polymer decreases as increases, leading to a decrease in te entropy. Suc a force versus extension curve F( ) for a force applied in te -direction ad been directly measured for a 97-kb λ-page DNA dimer []. Te experimental data can be fitted remarkably well using a worm-like cain (WLC) polymer model []. However, in te WLC model, excluded volume effect was not taken into account. For a polymer in solution, excluded volume effect is always present, except peraps at te tricritical θ-point. Terefore tere is in general no reason for tis effect to be negligible for DNA in solution. In tis paper we investigate te effect of excluded volume in DNA stretcing. We find tat for a week force F applied at te ends of te molecule, te extension in te direction of te force is not linear in F, but rater proportional to F γ, wit γ(-ν)/ν, were ν is te well-known universal correlation lengt exponent. Using te result ν0.6, [3] tis gives γ/3. We find tat te experimental data for λ-page DNA at small F is actually in agreement wit tis beavior wit γ 0.64. Including te excluded volume effect in a freely joint cain (FJC) model by coosing te segment lengt to reproduce te small extension beavior gives excellent fit to te full range of experimental data.
In two dimensions, te exponent ν is exactly known to be ¾[3]. Tis will give an extension proportional to F /3 in a week force F. In te FJC model, te functional form of te force versus extension curve F( ) is different from tat of te tree dimensional case. Again te excluded volume effects can be included by coosing te segment lengt in te FJC to reproduce te small extension beavior. Tis suggests experiments to be performed in two-dimensional geometry to test te stronger effect of excluded volume in lower dimensions. In section II we will review te models in wic excluded volume effect is neglected, including te Gaussian cain and te FJC model. In section III we will discuss DNA model wit te excluded volume effect and use it to fit te experimental data. In section IV we will discuss te two dimensional case. Section V is te conclusion. II. MODELS WITHOUT EXCLUDED VOLUME EFFECT A. GAUSSIAN CHAIN In te Gaussian cain model, a long DNA molecule at equilibrium in solution as random walk statistics. Te probability distribution p() of its end-to-end distance is approximately Gaussian [4]: p ) π 3 / 3 3 exp, < R > < R > ( () were <R> is te average end-to-end distance. For fixed, te entropy of te DNA is Sk B log[p()], were k B is te Boltmann s constant. For a fixed end-to-end distance, te force in te -direction is given by 3
F S T 3 k / < R > () For a polymer of wit a contour lengt L, te persistence lengt P can be defined in te Gaussian cain as P<R> /(L) [4]. Tis persistence lengt is independent of te contour lengt since for a Gaussian cain, <R> L. In terms of te persistence lengt P, equation () can be written as LP 3k T B F. (3) Te extension is linear in te force, so te DNA beaves as a Hookean spring wit ero natural lengt and a temperature dependent effective spring constant. B. FREELY JOINT CHAIN In te freely joint model [5], te cain of contour lengt L consists of (L/b) freely joint segments eac of lengt b. Wen tis cain is stretced at its two ends wit a force F, eac segment is pulled at bot ends wit tis force F and tus tends to align it in te direction of F, wile termal fluctuations tend to orient it in random directions. For a force in te direction and denoting te angle between F and a particular segment as θ, te partition function of te cain can be written as Z π bf cos( θ ) exp sin( ) d k π θ θ 0 L / b 4πk bf sin bf k L / b (4) Te average extension of te cain in te direction is given by k B T log Z F bf k Lcot k bf bf LΛ, k (5) 4
5 were Λ(x)cot(x)-x - is te Langevin function. For small x, Λ(x) x/3. Terefore for small F, LbF/(3k B T). Comparison wit equation (3) gives bp. Substituting into equation (5) ten gives II. MODELS WITH EXCLUDED VOLUME EFFECT For a polymer wit excluded volume effect, te probability distribution of te end-to- end distance is given by [6] were A is a normaliation constant. Te exponent is related to te universal exponent for te average end-to-end distance by /(-ν) [7]. Te exponent α is strictly ero at te upper critical dimension d c 4, wen ν/ and te probability distribution reduces to te form given in equation (). We will sow in te following tat α sould be ero in all dimensions. Te entropy is given by Te force exerted at te ends of te cain is given by Λ T k FP L B ) (6, 3 exp ) ( > < > < α R R A p (7) > < > < + α R R A k p T k S B B 3 log log ) ( log (8) > < > < 3 α R R T k S T F B ) (9
Unless α0, te first term would imply te unpysical result tat te force F becomes infinite as te extension goes to ero. Putting α0 in equation (7), te component of te force in te direction is given by F S T 3 k ) / ( ) ( < R > (0) Te average force F can be obtained by taking x y : F < R > < R > 3 / k ( ) / 3 k (3 ) () Te extension of te cain in te direction due to a force F in te direction is given by F < R > / 3 k () Terefore wit excluded volume effect, te extension is not linear in F for. In Figure we sow te double logaritmic plot of te experimental data for te extension versus force of a 97-kb λ-page DNA dimer []. Te slope of te curve at small extension is about 0.64. Tis is in good agreement wit te exponent /(-)/3 obtained using te known value of ν0.6 [3]. Tis sows tat excluded volume effect is important for DNA. In te freely joint cain model we again assume a cain consisting of L/b freely joint segments eac of lengt b. Ten te average extension in te direction of te force is 6
again given by equation (5). Te effects of excluded volume can be included in te FJC model by coosing te segment lengt b to reproduce te small extension beavior. Comparing te small extension result of tat equation LbF/(3k B T) wit equation (), we obtain for te segment lengt b 3k < > F R / FL 3 k (3) Substituting tis into equation (5) we obtain te expression for te extension in te direction of te force for a cain wit excluded volume effect F < R > LΛ / L 3 k 3 (4) In contrast to te case of a Gaussian cain, te persistence lengt defined by P<R> /(L) is not a constant but would depend on te contour lengt L. Terefore we use ere te average end-to-end distance <R> as a parameter, rater tan te persistence lengt. We ave fitted te experimental data []to equation (4), wit /(-ν).5, but varying te parameters L and <R> to minimie te least square deviation. Te best fit is obtained wit <R>3.9µm and L3.8µm. In Figure we sow te result of our fit wit te experimental data of te extension versus force. Te dased lines are te results of te freely joint cain wit no excluded volume effect. Te value P3nm corresponds to best least square fit wit te data and P50nm corresponds to a fit to te small extension data. We see tat te fit to te experimental data is excellent for te model wit excluded volume effect. A least square fit varying all tree parameters <R>, L and yields.6. Tis sows tat te form /(-ν) is in fact very good. 7
We ave also fitted te more recent data of Wang et al [8 ] to equation (4). Teir experimental data were obtained using a double stranded DNA of muc sorter sie (~µm). Wang et al. found tat teir data could be fitted well wit te teoretical forms of Marko and Siggia [9], (wit persistence lengt P43.5nm, contour lengt L.3µm) and Odjik [0] (wit persistence lengt P4.nm, contour lengt L.3µm). Bot forms take into account also elasticity teory. However, tey found tat teir experimental data were not well fit by te freely joint cain expression over any force region. Tis finding in fact supports conclusions of previous work wit double-stranded DNA molecules. Te freely joint cain model, owever can be successfully applied to describe singly stranded DNA. Since our teory wit excluded volume is based on te freely joint cain model we do not expect it to fit well te data of a double stranded DNA. In spite of tis we sow in Figure 3 our best fit to te data of Wang et al. [8]. Te dased curve is obtained using te freely joint cain model witout excluded volume, wit persistence lengt P4nm and contour lengt L.3µm. Te solid curve is te best fit using our expression (4) wit <R>0.394µm, L.3µm and.5. Even toug bot te freely joint cain and our equation (4) do not fit te data well we find tat our expression seem to fit better over most of te data range. We also find tat te largest slope in te data is about 0.5. Terefore even toug te force-extension relation is nonlinear, te exponent is muc smaller tan te value /3 predicted by our teory. Tis is again due to te double stranded nature of te DNA. It sould be pointed out tat our expression (4) applies only wen te cain is not yet close to fully extended. Wen te cain is close to fully extended, te freely joint cain expression (6) sould be better, because ten te excluded volume effect must be 8
negligible. One could, in principle tink of a way of bridging te two expressions (6) and (4). However, it is not known at wat extension, or force, is te excluded volume effect in te cain negligible. Only wen te cain is fully extended it is clear tat te excluded volume effect is negligible. But te fully extended configuration is only one among a large number of configurations wose number increases exponentially wit te lengt of te cain. For a cain tat is close to but not yet fully extended te critical force for te crossover can only be determined by comparing te fit of te experimental data to (6) and (4) to see at wat extension is te expression (6) a better fit tan (4). Looking at Figure, one can see tat te fits to te data from (6) and (4) are almost indistinguisable. Terefore we prefer to leave (4) as it is, keeping tis condition in mind. III. DNA IN TWO DIMENSIONS A. WITHOUT EXCLUDED VOLUME EFFECT Te partition function for a freely joint cain given in equation (4) is valid only for tree dimensions. In two dimensions it as to be replaced by Z L / b π Fbcosθ Fb dθ exp I 0 k π k 0 L / b (5) were I 0 is te modified Bessel function of order ero. Te average extension in te direction of te force is given by k Fb Fb T log Z L I I (6) F k k B 0 9
were I is te modified Bessel function of order one. From te beaviors of I 0 and I for small arguments one obtains LFb, F 0 k T B (7) Comparing tis to te Gaussian cain result for te two dimensional case, PLF /(k B T), we find bp. Substituting tis into equation (6), we find Fb L I k 0 Fb I k (8) for te extension of a two-dimensional DNA in direction of te force F. B. WITH EXCLUDED VOLUME EFFECT To take into account excluded volume effect we use equation (0) wit x + and find F k < R > ( ) k < R > (9) From tis we find ( ) / F < R > k (0) In a freely joint cain model wit (L/b) segments, eac of lengt b, te extension in te direction of te force is given by (6), wit te small F beavior given by equation (7). Again te excluded volume effect can be included by coosing te segment lengt in te FJC model to reproduce te small extension beavior. From equations (7) and 0
(0) we find b ( ) / k T F < R > B () LF k T B Substituting tis value of b into equation (6) we ave ( ) / F < R > LI L k ( ) / F < R > I0 L k () In two dimensions, te exponent for te end-to-end distance is known exactly to be ν3/4 [3]. Tis gives /(-ν)4, a muc larger deviation from te value in te case of no excluded volume effect. Only two parameters L and <R> in equation () are left to fit experimental data. However since no experimental data is known for DNA in two dimensions, we will use te same values of L and <R> obtained in tree dimensions. Te results are sown in Figure 4 for bot wit and witout excluded volume effect. We suggest experiments to be performed in two dimensions to test te stronger effects of excluded volume in lower dimensions. IV. CONCLUSION We investigate te effects excluded volume on te stretcing of a single DNA in solution. We find tat for small force F, te extension is not linear in F but proportion to F γ, wit γ(-ν)/ν, were ν is te well-known universal correlation lengt exponent. A freely joint cain model wit te segment lengt cosen to reproduce te small extension beavior gives excellent fit to te experimental data of λ-page DNA over te wole
experimental range. Te parameters used in te fitting are te contour lengt L and te average end-to-end distance <R>. Te best fit is obtained wit <R>3.9µm and L3.8µm. Te fit to te experimental data is just as good as te worm like cain model [], in wic only te elastic energy is taken into account by treating te cain as a uniform elastic rod. It is possible tat bot effects are present in DNA. We sow tat excluded volume effects are stronger in two dimensions and also derive results in two dimensions wic are different from te tree dimensional results. Tis suggests experiments to be performed in tese lower dimensions Acknowledgement: Tis work as been supported by te Department of Energy grant DE-FG0-97ER5343 REFERENCES [] S. B. Smit, L. Fini, C. Bustemante, Science 58, (99) [] C. Bustamante, J.F. Marko, E.D. Siggia and S. Smit, Science 65, 599 (994) [3] H.E. Stanley, Introduction to Pase Transition and Critical Penomena, Oxford University Press (997) [4] R. H. Austin, J.P. Brody, E.C. Cox, T. Duke and W. Volkmut, Pysics Today, February, 3 (997) [5] A. Yu. Grosberg and A.R. Koklov, Statistical Pysics of Macromolecules (AIP Press, New York, 994); J.M. Scurr, S.B. Smit, Biopolymers 9, 6 (990) [6] C. Domb, J. Gillis and G. Wilmer, Proc. Pys. Soc. (London) 85, 65 (965) [7] M.E. Fiser, J. Cem. Pys. 44, 66 (966) [8] M. Wang et al. Biopys. J. 7, 335 (997) [9] J.F. Marko and E.D. Siggia, Macromolecules 8, 8759 (995) [0] T. Odjik, Macromolecules 8, 706 (995)
FIGURE CAPTIONS Figure : Double logaritmic plot of extension versus force for λ-page DNA. Te slope of te curve at small extension is about 0.64 Figure : Extension versus force. Te solid dots are experimental data of λ-page DNA. Te solid line is for freely-joint cain wit excluded volume effects, using te best least square fit values for <R> and L and fixed value of.5. Te dased curves are results of freely joint cain, for two values of te persistence lengt. Figure 3: Extension versus force. Te solid dots are experimental data of ref. [8]. Te solid line is for free-joint cain wit excluded volume effects, using te best least square fit values of <R> and L and fixed value of.5. Te dased curve is te result of te freely joint cain. Figure 4: Extension versus force for a two dimensional polymer in te freely joint cain model. Te dased curve is for te freely joint cain model in two dimensions witout excluded volume, wit persistence lengt P50 nm. Te dotted curve is te same ting for tree dimensions, for comparison. Te solid curve is for te freely joint cain model in two dimensions wit excluded volume, using parameter values of <R> and L obtained in tree dimensions. 3
EXTENSION( µ m) 0 slope0.64 0.0 0. 0 FORCE(pN) Figure 4
35 30 P50(nm) EXTENSION( µ m) 5 0 5 0 5 P3(nm) <R>3.9µm L3.8µm.5 0 0.0 0. 0 00 FORCE(pN) Figure 5
.5 extension( µm) P4(nm) slope0.5 <R>0.394(µm) L.3(µm).5 0.9 0 force(pn) Figure 3 6
35 30 Gaussian EXTENSION( µ m) 5 0 5 0 dimensions Gaussian 3 dimensions excluded volume 5 dimensions 0 0.0 0. 0 00 FORCE(pN) Figure 4 7