# Entropic elasticity of polymers and their networks

Contents 1 Introduction 1 I Heterogeneous Freely-Jointed Chain Model and Its Applications 13 2 Mechanics of Forced Unfolding of Proteins 14 2.1 Introduction............................14 2.2 Three Equations Governing the Forced Unfolding of Proteins.16 2.3 Heterogeneous FJCModel and the EquilibriumForce-Extension Relation..............................16 2.4 Kinetic Equation.........................20 2.5 Constant Velocity Pulling I – Forced Unfolding of Globular Pro- teins................................22 2.6 Constant Velocity Pulling II – Forced Unfolding of Fibrous Pro- teins................................24 2.7 Pulling with a Force Linearly Increasing with Time......26 2.8 Constant Force Pulling......................29 2.9 Discussions............................32 2.10 Conclusions............................35 II Heterogeneous Wormlike Chain Model and Its Ap- plications 40 3 Statistical Mechanics of a Discrete Systemwith Quadratic En- ergy 41 3.1 General Theory..........................42 3.2 Statistical Mechanics of a Constrained System:Method 1...43 3.3 Statistical Mechanics of a Constrained System:Method 2...45 3.4 An Example............................45 4 Heterogeneous Wormlike Chain Under End-to-end Force 49 4.1 Introduction............................49 4.2 Description of the Heterogeneous Wormlike Chain Model...51 v

4.3 General Theory..........................53 4.4 Hinged-hinged 2D chain.....................57 4.5 Partially Clamped 2D Chain...................60 4.6 Clamped-clamped 2D Chain...................61 4.7 Fluctuation of a 2D Chain....................63 4.8 Theory for the 3D Chains....................64 4.9 Monte Carlo Simulation.....................66 4.10 Results and Application.....................66 4.10.1 Thermo-mechanical properties of the chain.......66 4.10.2 Fluctuation and correlation of the angle θ i .......67 4.10.3 Transverse ﬂuctuation of the chains y 2 ........68 4.10.4 Application to the protein unfolding problem......68 4.11 Conclusions............................69 5 Statistics of the Heterogeneous Wormlike Chain 78 5.1 Distribution of the End-to-end Extension P(x).........78 5.2 Distribution of the Transverse Displacement..........84 5.3 Conclusions............................85 6 Fluctuating Elastic Filaments Under Distributed Loads 88 6.1 Introduction............................88 6.2 Theory...............................90 6.2.1 Theory for a continuous elastic ﬁlament.........90 6.2.2 Energy of a discretized elastic ﬁlament or semi-ﬂexible chain............................94 6.2.3 Partition function and free energy............96 6.2.4 Force-extension relation.................97 6.2.5 Thermal ﬂuctuation around the ground state......97 6.3 Results...............................98 6.4 Conclusions............................100 7 Transition Between Two Regimes Describing Internal Fluctu- ation of DNA in a Nanochannel 105 7.1 Introduction............................106 7.2 Results and Discussion......................108 7.3 Materials and Methods......................119 7.3.1 Sequence speciﬁc labeling and DNA staining......119 7.3.2 Loading DNA into nanochannels............119 7.3.3 Microscopy and image processing............120 7.3.4 Recording and calculations................120 7.3.5 Partition function and angle ﬂuctuation........120 8 Entropically Driven Motion of Polymers in Non-uniformNanochan- nels 124 8.1 Introduction............................124 vi

8.2 Entropically Driven Diﬀusion..................126 8.3 DNA Conﬁned in Non-uniform Channels – Theory and Com- putation..............................129 8.4 DNA Conﬁned in Non-uniform Channels – Results......134 8.4.1 Stationary DNA in nanochannels............134 8.4.2 Migration and deformation of DNAin non-uniformchan- nels.............................135 8.4.3 Transition to the de Gennes regime under non-zero force 138 8.5 Conclusions............................139 III Statistical Mechanics of Filamentous Networks 148 9 Entropic Elasticity of Fluctuating Filament Networks 149 9.1 Introduction............................150 9.2 Expansion of a Triangular Network...............152 9.3 Entropic Elasticity of a General 2D Network – Theory....155 9.3.1 Stable equilibrium states.................156 9.3.2 Thermal ﬂuctuations around the static state......158 9.4 Entropic Elasticity of a Hexagon.................163 9.4.1 Hydrostatic edge tension on a hexagon.........163 9.4.2 Simple shear on a hexagon................171 9.4.3 Pure tension on a hexagon................181 9.5 A Comparison with Other Networks...............186 9.6 Conclusions............................186 10 Conclusions and Future Work 191 Appendices...............................193 A Evaluating the Partition Function 193 B det M for the Hinged-hinged Chain 194 C Force-extension Relation for a Homogeneous Wormlike Chain196 D Force-extension Relation for a Special Heterogeneous Worm- like Chain 198 E det M for the Clamped-clamped Chain 200 F Transverse Fluctuation Scales as T 202 G Partition Function for a Fixed Extension Ensemble 203 H Theory for the 2D Chains 205 vii

I Relation Between the End-to-end Extension x and Channel Width D for Wang and Gao’s Theory 207 J Fluctuation for Short Internal Segments 208 K Heterogeneity on the Backbone of DNA 210 L Total Extension versus L Relation 212 M Distribution of Extension in the Deﬂection Regime 214 N Results of Entropy-induced Migration Derived fromthe Sackur- Tetrode Equation 215 O Transverse Size of a Strongly Conﬁned Polymer 217 viii

List of Tables 2.1 Eq.2.39 divides the F − t plane into 4 regions with diﬀerent signs of dF/dt...........................34 ix

List of Figures 1.1 Beam structures made by human beings and by nature.(A) The Akashi-Kaiky¯o Bridge,located in Japan and spanning 1,991 metres,is the world’s longest suspension bridge (Figure comes from http://blogger.sanook.com/confuse/2008/11/23/).(B) In a cell division cycle,the mitotic spindels,composed by micro- tubules and various other proteins,are used as ‘cables’ to pull the daughter chromosomes apart.(Figure comes from reference [1].) (C) Man-made tent is usually supported by a frame of poles.(Figure comes from Wikipedia:Tent.) (D) A spectrin meshwork under the human red blood cells.Cell membrane is thin and fragile.So most of them are strengthened and sup- ported by a network of protein ﬁlamenets.This ﬁlamentous network,like the frame in the man-made tent,determines the shape of the membrane.(Figure comes from reference [1].) (E) The 2010 Vancouver Olympic cauldron.(Figure comes from Wikipedia:Olympic Flame.(F) Structure of the stereocilia projecting from a hair cell in the inner ear (Figure comes from [2]).................................3 1.2 Force-extension relation for a wormlike chain.In the small force limit (red dashed line,Eq.1.10),the polymer behaves like a spring with a temperature dependent spring constant.In the high force limit (black dashed line,Eq.1.14),the model pre- dicts that 1 −x/L ∼ F −1/2 .These two limits can be combined to construct an approximate formula for all forces (blue line, Eq.1.15)..............................5 x

1.3 Set-ups for measuring the force vs.extension behavior of macro- molecules (Figure comes fromreference [16]).(A) Hinged-hinged condition.Both ends are attached to beads which are held in optical traps that can exert forces but not moments.Hence the curvatures at the ends are constrained but not the slopes.The trap does not allow transverse displacements.(B) Clamped- clamped condition.The slopes and transverse displacements are constrained at both ends.(C) Partially clamped condition. One end of the macromolecule is secured to a cover-slip while the other end is attached to a bead in a magnetic or optical trap which ensures that the slope is held constant,but trans- verse displacements are allowed..................9 2.1 Illustration of the two-state kinetic model.(a) A chain of mixed folded and unfolded proteins is modeled as a heterogeneous freely-jointed chain.A single folded (unfolded) protein is repre- sented by an N fs -segment (N us -segment) subchain with Kuhn length l f (l u ).In this illustration,two folded and one unfolded proteins are represented by the two red and one blue subchains respectively.Note that in reality,the actual number of segments in each subchain may be much larger.Also,l u is expected to be smaller than l f since an unfolded protein is expected to be ﬂoppier than a folded protein.(b) Energy landscape of the two-state model.The ordinate is the Gibbs free energy and the abscissa is the reaction coordinate.The two wells,representing the folded and unfolded states of a protein,are separated by an energy barrier with transition distances ∆x u and ∆x f .At zero force,the folding rate and the unfolding rate are β 0 and α 0 respectively,with β 0 >> α 0 .An applied force can lower the energy barrier and thus change the folding and unfolding rates.19 2.2 Predictions of the force-extension proﬁles using the heteroge- neous FJC model with only four free parameters.(a) ubiquitin. (b) ﬁbrinogen.Blue curves are the experimental data – ubiqui- tin data from [7] and ﬁbrinogen data from [9].We use two of the experimental curves (black dots) to ﬁt the Kuhn length and the contour length of the folded and unfolded proteins (black dashed lines are the ﬁtting results).Then,without any more free parameters,we use the heterogeneous FJCmodel (Eq.2.12) to predict all the other curves.The predictions (red curves) match well with the experimental data for both proteins....21 2.3 Force-extension proﬁles of six copies of (a) ubiquitin,and (b) ﬁbrinogen (v c = 1000nm/s) in constant velocity pulling.Blue curves:experimental data (ubiquitin data from [7] and ﬁbrino- gen data from [9]);red curves:prediction using our two-state kinetic model............................23 xi

2.4 Force-extension relation (continuous model) of ubiquitin (red) and ﬁbrinogen (blue).In reality,both are globular proteins. Here we plot the force-extension relation using their kinetic pa- rameters,but with N f ,N u being real numbers instead of integers.26 2.5 Number of folded proteins as a function of time (continuous model).Red curve for ubiquitin and blue curve for ﬁbrinogen. In reality,both are globular proteins.Here we plot the N f (t) proﬁle using their kinetic parameters,but with N f ,N u being real numbers instead of integers..................27 2.6 Constant velocity pulling.(a) Dependence of the average un- folding force on pulling speed.Red:ubiquitin;blue:ﬁbrinogen; solid line:accounting for refolding (k f = 0),dashed line:ig- noring refolding (k f = 0).For both proteins,the predicted unfolding force using the set of parameters that assume k f = 0 is much smaller,especially at high pulling velocities,than the one that takes the refolding rate into account.(b) Relaxation proﬁles (v c = 0nm/s).The evolution of the number of refolded proteins is shown as a function of the relaxation extension x r (normalized by x max = NL us ).A limit proﬁle (blue) is ap- proached as time approaches inﬁnity...............28 2.7 Extension versus time proﬁles for pulling using a linearly in- creasing force (v f = 300pN/s,9 copies of proteins).(a) ubiqui- tin and (b) ﬁbrinogen.Red curves:predicted step-wise proﬁle, using integer N f and N u in the governing equations (Eq.2.12 and Eq.2.13).Each of the steps obeys the equilibrium force- extension relation,which is shown as dashed black curves (Eq.2.12). Red circles and dashed line:unfolding events and the ﬁtting re- sults.The ﬁtting equation is x = 680.9t −52.12 for ubiquitin and x = 1419t − 92.01 for ﬁbrinogen (x is the extension in units of nm.Blue lines:solutions obtained by using continuous N f and N u in the governing equations,which match well with the discrete model shown in red.The proﬁles for ubiquitin ob- tained here are consistent with those measured in [8].Insets:if we take the refolding rate into account,some refolding events are observed at small force.(c) Extension versus time proﬁles for nine copies of ubiquitin assuming k f = 0 shows no refolding events at small force even though it reproduces the overall trend.28 xii

2.8 The double exponential function W(F) together with the sta- tionary points (blue) and the critical forces (red) for pulling with a force linearly increasing in time.Assume that there are nine copies of ubiquitin in the chain.The double exponential function when N f = 8,N u = 1 is shown as a black solid line (when N u = 0,the function is only a single exponential).As more and more proteins unfold,N f decreases and the curve W(F) shifts to the right along the F axis (see the arrow).We plot W(F) in black dashed line for N f = 5,N u = 4 and in black dashed-dotted line for N f = 1,N u = 8.The movement of the curve results in the increase of the critical force F cl (red)....29 2.9 Constant force pulling proﬁles.(a) Extension versus time proﬁle for nine copies of ubiquitin.Blue:F = 100pN,red:F = 120pN, black:F = 140pN.Inset:F = 80pN.Step-wise curves:solu- tion when assuming N f and N u to be integers in both Eq.2.12 and Eq.2.13.Continuous curves:analytic solution obtained by letting N f and N u be real numbers in the equations.The dwell time for the step-wise solution,which is obtained analytically in the text,increases as the protein unfolds.The inset shows that when the force is not large enough,not all the proteins can unfold,and refolding/unfolding ‘hoppings’ occur periodically. (b) N f∞ /N as a function of the dimensionless force Π F .Red: ubiquitin;Blue:ﬁbrinogen.δF,the range of the dimensionless force over which N f∞ /N changes from 90% to 10%,is a uni- versal constant ln81 for all proteins.Π ∗ F is the dimensionless force for half the proteins to unfold.It is shown in the text that Π ∗ F = ln(β 0 /α 0 ).(c) Unfolding rate α as a function of the applied force F.Red solid line is the result of taking the refold- ing rate into account.When the force is large enough,log α is a linear function of F (red broken line).Blue solid line is the prediction assuming k f = 0.Inset,k f /k u as a function of the applied force............................31 3.1 A simple coupled harmonic spring system with two balls at- tached to three springs.All three springs have the same spring constant k and natural length L.The degrees of freedom for this discrete system are two,characterized by x 1 and x 2 ,the displacements of the two balls away from their equilibrium po- sitions................................46 xiii

4.1 Model of the 2D chain.A thermally ﬂuctuating N−segment 2D chain is subjected to an external applied force

F = F ˆ X.The conﬁguration of the chain is characterized by its N tangent an- gles θ i ,formed by the segments with respect to the ˆ X axis.The transverse displacement of the chain,denoted by y i in the ﬁg- ure,reﬂects how much the chain ﬂuctuates.(a) Hinged-hinged boundary conditions:both ends of the chain are constrained on the ˆ X axis,but no moment is acting on them;(b) partially clamped boundary conditions:one end of the chain is clamped on the ˆ X aixs while the other end,with slope also constrained to be zero,is free to have transverse displacement in the Y di- rection.(c) clamped-clamped boundary conditions:both ends are clamped on the ˆ X axis....................51 4.2 The unkown function W(F) can be measured in a single force extension experiment.The shaded area above the force-extension curve is the complementary energy and the area beneath the force-extension curve is k B T · W(F) by Eq.4.15.........55 4.3 Thermo-mechanical quantities for a ﬂuctuating chain.Blue: hinged-hinged boundary conditions;red:partially clamped bound- ary conditions;black:clamped-clamped boundary conditions. (a) Force-extension proﬁle of the chain.Inset:local proﬁle shows that the hinged-hinged chain (blue) has smaller exten- sion and thus is more ﬂexible (to show the ﬁgure clearly,we have changed the circles into lines with the same colors);(b) Variance of the extension.Inset:local proﬁle shows that the hinged-hinged chain (blue) ﬂuctuates more than the partially clamped chain (to show the ﬁgure clearly,we have changed the circles into lines with the same colors);(c) Average energy of the chain versus the applied force;(d) Variance of the energy; (e) Thermal expansion coeﬃcient α versus the applied force;(f) Isothermal extensibility χ versus the applied force........70 4.4 Verifying Eq.4.23.Solid line:theoretical prediciton;circles: MC simulation results.Simulations have been done under 11 diﬀerent forces varying from150pN to 1150pN with an increase- ment of 100pN.Temperature is set to be 300K.Relative exten- sion as well as thermal expansion coeﬃcient are recorded.The result shown is for a homogeneous chain with contour length L = 25nm and bending modulus K = 2.5k B T · nm.......71 xiv

4.5 Force-extension relation for homogeneous chains (blue curve) and rods (red circle,theory in [17]).(a) Hinged-hinged bound- ary conditions;(b) partially clamped boundary conditions;(c) clamped-clamped boundary conditions.K = 2.5k B T · nm, L = 2.5nm.The ﬁgures show that our force-extension relations for the chains reduce to the known formulae for the continuous rods when N → +∞ with L = Nl ﬁxed.Here N = 50000 for the blue curves...........................71 4.6 Dependence of the ﬂuctuation of θ angles on the boundary con- ditions.Blue:hinged-hinged chain,the ﬂuctuation in θ is at maximum and minimum respectively at the two ends and in the middle of the chain;red:partially clamped chain,the ﬂuc- tuation is at maximumand minimumrespectively in the middle and at the two ends;black:clamped-clamped chain,the ﬂuc- tuation is at minimum at the two ends,but the maximum is not achieved in the middle of the chain.Also,the partially clamped chain (red) ﬂuctuates more than the clamped-clamped chain (black).(a) ξ p /L = 5,the dependence on the boundary conditions is signiﬁcant througout the chain;(b) ξ p /L = 0.2 the dependence on the boundary conditions is signiﬁcant only at the two ends of the chain.To make the ﬁgures clear,the MC simulation results are not shown in the same ﬁgures......72 4.7 Dependence of the ﬂuctuation of θ angles on the heterogeneity of the chain.Blue:homogeneous chain with K = 2.5k B T · nm; black:corresponding MCsimulation results;red:heterogeneous chain with two bending moduli:K I = 0.5k B T · nm at the ﬁrst half of the chain and K II = 4.5k B T·nmat the second half;black dashed curve:corresponding MC simulation results.(a),(b) and (c) are for hinged-hinged,partially clamped and clamped- clamped boundary conditions respectively.The ﬁgures show that jumps in the bending modulus result in jumps in the ﬂuc- tuation in the θ 2 proﬁle.The larger the bending modulus,the smaller the ﬂuctuation in θ....................72 xv

4.8 Correlation in the tangent angle θ.(a-f):results for the ho- mogeneous chains.Blue:ξ p /L = 5;red:ξ p /L = 1;black: ξ p /L = 0.2.(a),(c),(e) are the theoretical results for the hinged- hinged,partially clamped and clamped-clamped chains respec- tively.To make the plots clear,we plot the corresponding MC simulation results separately in (b),(d) and (f) (circles). The ﬁgures show that the correlation in θ depends strongly on ξ p /L.When ξ p /L > 1 (blue),the proﬁle also signiﬁcantly depends on the boundary conditions.(g-h):results for a het- erogeneous chain with L = 1nm.The ﬁrst half and the second half of the chain have bending moduli of K I = 0.5k B T · nm and K II = 4.5k B T · nm respectively.(g) is the theoretical pre- dictions and (h) is the MC simulation results.Blue,red and black colors are for the hinged-hinged,partially clamped and clamped-clamped boundary conditions respectively.The corre- lation proﬁle loses its symmetry and decreases faster at the ﬁrst half of the chain where the bending modulus is smaller.....73 4.9 Transverse ﬂuctuation y 2 .(a):blue:hinged-hinged chain; red:partially clamped chain;black:clamped-clamped chain. Solid curve:homogeneous chain with K = 2.5k B T · nm;dashed curve:heterogeneous chain with K I = 0.5k B T · nm for the ﬁrst half of the chain and K II = 4.5k B T · nm for the second half.In (a),L = 1nm for all the curves.Since the curves are close to each other,to make the theoretical results clear,we do not plot the MC simulation results in (a).(b):transverse ﬂuctuation decreases when the force increases.The results are for a homogeneous hinged-hinged chain with L = 25nm and K = 2.5k B T · nm.The corresponding forces are labeled in the ﬁgure.Circles:MC simulation results;solid lines:theoretical predictions.............................74 4.10 Dependence of the transverse ﬂuctuation on the countour length of the chain.K = 2.5k B T · nm for all the curves.Black solid curves (theory) and blue circles (MC simulation):L = 5nm; black dashed curves (theory) and red circles (MC simulation): L = 25nm;(a) hinged-hinged boundary conditions;(b) partially clamped boundary conditions;(c) clamped-clamped boundary conditions.The ﬁgures show that for a ﬁxed persistence length, the longer the chain,the more the ﬂuctuation.Also,our theo- retical results and the MC simulation results match quite well. Note that here ξ p /L ≤ 1 and the results for hinged-hinged chain and clamped-clamped chains are quite similar,which is con- ﬁrmed by the simulation results..................74 xvi

4.11 Unfolding of six copies of ubiquitins under constant velocity pulling condition.Blue dotted curves are the experimental data from [25].Each peak in the proﬁle represents a unfolding event where the force drops.The ﬁrst and the last experimental curves are ﬁtted to obtain the contour lengths and the bend- ing moduli of the folded and unfolded proteins (Fig.(a):red circles are the ﬁtted data and the black curves are the ﬁtting results).The intermediate curves are then predicted without any free parameters using the 3D version of Eq.4.38 (Fig.(b), red curves).Figure.(b) shows that the predictions match well with the experimental data....................75 5.1 End-to-end distance distribution function P(x) using Eq.5.14 (red circles) and Eq.5.19 (blue).Here the external force is F = 0pN,the segment length is l = 1nmand the contour length is L = 50nm.The two theories match well with each other...82 5.2 (A):End-to-end distance distribution function P(x) for a freely- jointed chain with diﬀerent contour lengths L = 5,10,25nm. Here F = 10pN and the Kuhn length is l = 5nm.The proﬁle looks symmetric only when the contour length of the chain is long compared to its Kuhn length.(B) Distribution P(x) for a freely-jointed chain under diﬀerent values of tensile forces: F = 5,10,20pN.As the force increases,the proﬁle shifts to the right and the peak becomes sharper.For both plots,T = 300K.83 5.3 (A):End-to-end distance distribution function P(x) for a worm- like chain with diﬀerent contour lengths L = 1,5,25nm.Here F = 1000pN.For short chain,the proﬁle is clearly not sym- metric.When the contour length of the chain increases and becomes comparable to its persistence length,the proﬁle looks more symmetric.(B) Distribution P(x) for a wormlike chain under diﬀerent values of tensile forces:F = 50,100,300pN.As the force increases,the proﬁle shifts to the right and the peak becomes sharper.For both plots,T = 300K,K b = 2.5k B T· nm.83 5.4 Distribution P(r,x) for (A) a freely-jointed chain and (B) a wormlike chain.The chains are subjected to hinged-free bound- ary conditions.x,r are respectively the extension and trans- verse displacement of the free end.Here x is ﬁxed for each curve and P(r) versus r is plotted.The parameters are (1) Tempera- ture T = 300K;(2) Kuhn length l = 1nm (freely-jointed chain) or bending modulus K b = 2.0k B T (wormlike chain);(3) Con- tour length L = 25 nm;(4) Fixed force F = 50pN.The ﬁgures show that as the ﬁxed x increases,the peak in the distribution proﬁle becomes sharper.This makes sense because a chain with large extension has less freedom to ﬂuctuate in the transverse direction..............................86 xvii

6.1 A ﬂuctuating elastic ﬁlament (extensible wormlike chain) under distributed forces.The origin of the x −y coordinate system is set at the head of the ﬁlament,which is hinged.The other end of the ﬁlament is constrained to move only in the x direction. One possible deformed conﬁguration of the ﬁlament is shown in dashed line.............................90 6.2 Comparison between the continuous models and the discrete model.(A) Force balance for an inﬁnitesimal segment of a con- tinuous rod.(B) Comparison of the results for a continuous rod (Black curve:Fourier series method and Eq.6.15;Blue (almost overlaps with the black curve):method using force balance on inﬁnitesimal segment and Eq.6.20) and a discrete chain (red cir- cles).The ﬁlament is under constant τ along the arc length so that Fourier series method can be applied.Here a 100nm chain is discretized into 1000 segments.The results match quite well.99 6.3 Force-extension relations for a wormlike chain (1:red solid line) under uniform distributed load τ with thermal ﬂuctuations,(2: red dashed line) under uniform distributed load τ without ther- mal ﬂuctuations,(3:blue solid line) under end-to-end force F = τL 0 with thermal ﬂuctuations,and (4:blue dashed line) under end-to-end force without thermal ﬂuctuations.The refer- ence contour length of the chain is L 0 = 50nm.The persistence length is 5nm.The segment length is 0.5nm with N = 100 segments..............................100 6.4 Transverse ﬂuctuation of a chain under uniformdistributed τ = 5pN/nm (red),and under end-to-end applied force F = τL 0 (blue).Under distributed force,the chain has larger thermal ﬂuctuations with an asymmetric ﬂuctuation proﬁle.......101 xviii

6.5 DNA in non-uniformmicroﬂuidic channels.(A) A piece of DNA conﬁned in a linear channel and a constant-strain rate channel. Both channel types have been fabricated in experiments [15]. (B) The velocity in the non-uniform channel is inversely pro- portional to the channel width.Therefore,given the velocity v f at the exit (rightmost) end,the entire velocity proﬁle in- side the channel is known,which then leads to the drag force τ = d t v along the polymer.Here the end-to-end extension of the polymer is plotted against v f .As we increase the ﬂow ve- locity,the strain along polymer increases,resulting in a larger end-to-end extension.Red:DNA in a linear channel.Blue: DNA in a constant-strain-rate channel.Dashed/Solid lines:ex- tension with/without the contribution of thermal ﬂuctuations. (C) Transverse ﬂuctuations along the polymer arc length.Red and blue for DNA in a linear and a constant-strain-rate channel respectively.Solid line is for a DNA with one end hinged and the other end free to ﬂuctuate.Dashed line is for the same DNA with both ends hinged on the x aixs...............102 6.6 Transverse ﬂuctuation of a chain under uniform distributed τ plus a point load F in the middle.The left half of the chain has less ﬂuctuation because the stretching of the point loads reduces the thermal ﬂuctuations.................102 7.1 Measurement of the ﬂuctuations of the internal segments of con- ﬁned DNA.(A) Image of a dye label (Alexa-546) on a DNA backbone (backbone not shown) with 80ms exposure time.(B) 2D surface plot of the raw image (intensity of the dye vs.the X Y coordinates).(C) Image of one T4 DNA fragment (∼ 36 microns) with backbone (red) and internal labels (green).(D) Time series (8 seconds) of the DNA showing the ﬂuctuations of backbone and internal labels.In (D),the red trace is the backbone and the green traces are the trajectories of internal dye labels..............................107 7.2 Internal ﬂuctuation of λ DNA conﬁned in a 80nm×130nmchan- nel.(A) The measured rms ﬂuctuation σ versus mean exten- sion x for the internal segments of the DNA agrees very well with de Genne’s theory with no ﬁtting parameters (red curve, Eq.7.4).(B) A linear σ 2 − x proﬁle conﬁrms the 0.5 power law of σ ∼ x 1/2 of the de Gennes’ theory.Note,however,that here we have maximum x 10µm.As shown in a subse- quent ﬁgure (Fig.7.4) and in the text,for longer polymer with a maximum x 10µm,the data deviates signiﬁcantly from de Gennes’ theory and even the 0.5 power law is lost......108 xix