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From single-molecule to organism-level biophysics

Broekmans, O.D.

2016

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Broekmans, O. D. (2016). From single-molecule to organism-level biophysics.

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3

DNA twist stability changes with

magnesium(2+) concentration

Abstract — To understand DNA elasticity at high forces (F > 30 pN), its helical nature must be taken into account, as a coupling between twist and stretch. The prevailing model, the worm-like chain, was previously extended to include this twist–stretch coupling. Motivated by DNA’s charged nature, and the known effects of ionic charges on its elasticity, we explore the impact of buffer ionic conditions on twist–stretch coupling. After developing a robust fitting approach for force–extension data, we find that DNA’s helical twist is stabilized at high concentrations of the magnesium divalent cation.

3.1 Introduction

Mechanical perturbations of the DNA double helix form a crucial step

during many of the cell’s life-sustaining processes. When proteins bind,

replicate, compact, and repair the genome, the DNA molecule is bent,

stretched, and twisted. A detailed understanding of DNA’s elastic response

to these perturbations is therefore a prerequisite for a deep quantitative

in-sight into the biology of the cell. The single-molecule techniques that have

been developed over the past two decades [1, 2] have greatly contributed

to this understanding: it is now routinely possible to directly manipulate

single molecules of DNA, and monitor their response to stretch and twist

under a wide variety of experimental conditions. One such technique is

the optical tweezers (Fig. 3.1, schematic), which can be used to accurately

measure the force response of DNA [3]. By modeling the corresponding

force–extension data, we can then not only improve upon our structural

understanding of DNA; it is also possible to infer the mechanisms of action

of DNA-binding proteins from the changes they induce in the data [4, 5].

Below mechanical loads of ∼ 30 pN, dsDNA’s force response is

accu-rately modeled by the extensible worm-like chain (eWLC) [6, 7]. This

well-established, semiclassical model describes the molecule as simply an

isotropic, extensible rod: Entropic bending fluctuations, characterized by

a persistence length L

(50 nm for dsDNA under physiological conditions),

and enthalpic stretching of the DNA backbone, characterized by the stretch

modulus S (1 500 pN [8]), are balanced by the work performed by the

stretching force F , leading to a relative extension d/L

(end-to-end

dis-tance over contour length). Beyond 30 pN, however, dsDNA’s helical

struc-ture needs to be taken into account [9, 10, 8]. This introduces an energy

cross-term between the molecule’s twist and stretch degrees of freedom.

Only relatively recently, Gross et al. found it possible to incorporate this

effect into the eWLC, yielding the twistable worm-like chain (tWLC) [8]:

3.1 Introduction

where C is the molecule’s twist rigidity (440 pN nm

[8]), and g(F ) is the

twist–stretch coupling. As illustrated schematically in Fig. 3.2a, a positive

value of g would correspond to the intuitive case of DNA unwinding as it

is being stretched. In reality, DNA slightly overwinds up to ∼ 35 pN — a

finding with major implications for proteins that have to twist DNA upon

binding [9, 10]. More importantly, g is force-dependent: above a critical

force F

of ∼ 31 pN, DNA gradually switches from overwinding to

under-winding [9, 8]. This is modeled in the tWLC by taking g as a piecewise

linear function of the force F (see Fig. 3.2b, and Methods). The

force-dependence leads to a marked deviation of the DNA’s force–extension

behavior from that of the extensible worm-like chain (cf. Fig. 3.1),

un-til, around 65 pN, the DNA undergoes a structural transition known as

overstretching, whereby the molecule suddenly lengthens by ∼ 70% over a

narrow force range [11, 12, 8, 13]. Overall, the tWLC substantially extends

the range of forces over which dsDNA force–extension data are understood

(see Fig. 3.1).

More importantly, the tWLC captures DNA twist–stretch coupling into

the two parameters g

and g

, which can now be obtained from fits to

force–extension data. This opens up the possibility of investigating if, and

to what extent, twist–stretch coupling is affected by buffer ionic

condi-tions. DNA is, after all, a highly charged molecule, known to interact

strongly with cations through its phosphate backbone and major groove

[14, 15]. These interactions can lead to softening of DNA [16, 17], and, for

multivalent cations, to bending [18], and even condensation [19]. In this

chapter, we therefore set out to quantify DNA twist–stretch coupling as a

function of the concentration of the divalent magnesium cation (Mg

).

3.2 Optical tweezers experiments

Using optical tweezers, we collected single-molecule force–extension data

on dsDNA at varying concentrations of magnesium (0–150 mM MgCl

). In

3.2 Optical tweezers experiments

30 40 50 60 70 For ce (pN) Distance (μm) 16.5 17 17.5 18

Figure 3.3 | Double-stranded DNA force–extension data for increasing con-centrations of MgCl2 (in a background of 500 mM NaCl and 10 mM TrisHCl, pH 7.6; light to dark: 0, 25, 50, and 100 mM of MgCl2, respectively). For high magne-sium concentrations, a clear stiffening of the DNA before the onset of overstretching is observed. Shown is data averaged per magnesium concentration (N ≥ 12), after correction for systematic measurement errors (see text).

3.05 µm streptavidin-coated polystyrene microspheres captured in two

op-tical traps, inside a microfluidic flowcell (for details on the instrument and

protocols, please refer to ref. 20). To suppress unpeeling of the untethered

ends of the DNA strands, which causes a signature saw-tooth pattern in

the overstretching plateau (see Fig. 3.1) [8], we worked in a background

of 500 mM monovalent salt (NaCl) [21, 13]. Below forces of 30 pN, force–

extension curves were indistinguishable (data not shown), indicating that

the persistence length L

and stretch modulus S were not affected by

3.3 Data analysis approach

3.3 Data analysis approach

To quantify the stiffening effect, we fitted the data with the tWLC model,

Eq. (3.1). Since previous reports have shown the twist rigidity C to be

insensitive to ionic strength [22, 23], that leaves the model with four fit

parameters: L

, S, g

, and g

. Given this large number of parameters, a

solid approach for fitting the data was needed. We would like to highlight

three key points in the approach we have developed: (1) fitting with the

force as the dependent variable; (2) correcting for systematic measurement

errors; and (3) global fitting with shared physical parameters. (For details,

see the Methods.)

The first point stems from the observation that, in optical tweezers data,

the force signal carries the most significant error (and not the distance,

which is precisely controlled). As such, when performing a least-squares

fit, the force should be the dependent variable [24]. This implies that the

model fitted to our data should be an inversion of Eq. (3.1), expressing

force as a function of distance. The impact of this inversion is illustrated

in Fig. 3.4, for the simplified case of an eWLC fit to simulated data. If

Eq. (3.1) is (incorrectly) used as-is for the least-squares fit, the value found

for L

changes wildly as more or less data from the low-force tail is included

in the fit — in addition to systematically underestimating L

. When,

instead, an inverted version of Eq. (3.1) is used, the fit result does become

reliable, both for L

and the stretch modulus S (see Methods, Fig. 3.6)

.

Second of all, we applied corrections for three systematic measurement

errors that are intrinsic to our data: (a) the force at zero extension is not set

exactly to zero during each experiment, leading to a random force offset F

for each force–extension curve; (b) small variations in microbead diameter

lead to a distance offset d

; and (c) imperfect force sensor calibration

causes force data to include a random factor δF . The first two systematic

errors were accounted for by including the offsets in the eWLC equation

0 20 40 60 80 100 0.6 0.7 0.8 0.9 Lp (nm) d_min / L_c dmin For ce discard fit Distance

3.4 Results

[see Methods, Eq. (3.5)]. This amended eWLC equation was fit to the

data below 30 pN, and the offsets found were subtracted from the data.

The third systematic error was rectified by using the force F

at which

the overstretching plateau occurs as a proxy for δF . Within the force

resolution of our instrument, there appears to be no correlation between

magnesium concentration and F

; we therefore rescaled all force–extension

curves to have overlapping overstretching plateaus.

Thirdly, and finally, we opted for a global fitting approach. We grouped

all force–extension curves into ensembles by magnesium concentration,

im-plying that the values of the physical parameters (i.e., L

, S, g

, and g

)

for curves within each ensemble should be equal. We could thus fit all

curves in each ensemble simultaneously, while sharing fit parameters

be-tween curves.

Fits of simulated data confirmed that, generally, global

fitting performs significantly better than individual fitting of the curves,

with a decreased sensitivity to the aforementioned systematic measurement

errors (see Methods).

As expected, the tWLC does not fit the full dsDNA force–extension

curve up until the overstretching plateau (Fig. 3.1): the onset of the

over-stretching phase transition needs to be excluded from the fit. We therefore

removed all force–extension data above a maximum force F

,

deter-mined by optimizing F

in each magnesium concentration ensemble for

a maximum coefficient of determination (R

) of the fit (see also Methods,

Fig. 3.10). This way, we were able to determine the tWLC fit parameters

for each of the measured magnesium concentrations.

3.4 Results

[Mg2+] (mM) 0 50 100 150 0 50 100 150 15 16 17 18 g1 (nm) -660 -640 -620 -600 -580 g0 (pN nm) 1.7 1.8 1.75 1.85 _{S (nN)} 40 41 42 43 44 Lp (nm)

Figure 3.5 | dsDNA twist stabilizes with increasing magnesium concentration. As the concentration of magnesium(2+) (MgCl2) is increased from 0 to 150 mM (in a background of 500 mM NaCl), the twist–stretch coupling parameter g1 decreases, indicating a stabilization of dsDNA twist (error bars: bootstrap error, N ≥ 12). In contrast, the persistence length Lp and stretch modulus S are relatively insensitive to these changes in divalent cation concentration.

well below the monovalent salt concentration (500 mM NaCl) used in our

buffer [16].

The values we find for the persistence length L

are slightly lower than

the widely accepted value of 50 nm at physiological salt concentrations.

At buffer conditions more similar to ours, however, previously reported

values are consistent [16, 17]. Similarly, we measure relatively high values

for the stretch modulus S. Both of these results are confirmed, however,

using alternative analysis approaches based on the eWLC (see Methods).

Our values for g

and g

at zero magnesium concentration are statistically

indistinguishable from previously reported values [8], and, for low forces,

are equivalent to values found in magnetic tweezers studies [9, 10].

In contrast to L

and S, g

shows an almost 15% decrease between 0–

alterna-3.5 Conclusions

tive fitting approach). This can be interpreted as a decreased tendency of

the DNA double helix to unwind under high tensile stress — in other words,

a stabilization of DNA twist. Qualitatively similar results have been shown

before in a bulk study of the relaxation of supercoiled, circular dsDNA by

topoisomerase I [25]. There, it was speculated that neutralization of the

DNA’s backbone charge by magnesium(2+) ions diminishes

intramolecu-lar repulsion, effectively stabilizing the helical twist of the molecule. The

subsequent inversion of the effect at still higher magnesium concentrations

is not observed in our study, however. This leaves open the question of the

exact molecular mechanism underlying the observed increase of twist

sta-bility, and whether the divalent salt homogeneously stabilizes DNA twist,

or specifically affects sequence-dependent melting transitions [26]. Future

experiments using fluorescent intercalators as reporters of DNA ligation

state may help answer these questions.

3.5 Conclusions

In conclusion, we have developed a robust analysis approach for force–

extension data, based on the twistable worm-like chain model. Our

ap-proach gives access to the elasticity regime between 30 and ∼ 60 pN, and

thus to information about twist–stretch coupling, directly from stretching

data. We have applied this technique to force–extension data of

double-stranded DNA at varying concentrations of the magnesium divalent cation

(0–150 mM, in a background of 500 mM NaCl). The observed stiffening of

the DNA at high magnesium concentrations can be interpreted as a nearly

15% decrease of the twist–stretch coupling parameter g

. More

3.6 Methods

3.6.1 Experimental procedures

The optical tweezers instrument used for this study has been described in detail elsewhere [20, 27]. In a microfluidic flowcell, two streptavidin-coated polystyrene microspheres (3.05 µm in diameter) were captured in two optical traps. λ-phage DNA (48.5 kb, Lc = 16.5 µm), which had been biotinylated using a custom pro-tocol [12], was tethered between the beads.

An initial force–extension curve was measured in a buffer solution consisting of 10 mM Tris-HCl (pH 7.6), and 500 mM NaCl. The background of monovalent salt ensured that unpeeling from the ends of the untethered strands was strongly disfavored [21]. After the initial force–extension curve, the molecule–bead as-sembly was moved into a separate channel of the flowcell containing the buffer plus a concentration of MgCl2. Here, after an additional calibration of the force response of the optical traps, another force–extension curve was recorded.

3.6.2 Data analysis approach

DNA elasticity models

In this chapter, the following two model equations for double-stranded DNA elasticity are referred to:

Extensible worm-like chain (eWLC) [6]: d Lc = 1 − 1 2 kBT F Lp !1/2 +F S (3.2)

Twistable worm-like chain (tWLC) [8]: d Lc = 1 − 1 2 kBT F Lp !1/2 + C −g(F )2_{+ SC} · F , (3.3) with g(F ) = ( g0+ g1Fc if F < Fc g0+ g1F if F ≥ Fc (3.4)

Simulated data

3.6 Methods

Fitting approach

In our optical tweezers data, the most significant error is in the force signal. Typically, force data has a 2% relative error (σF ∼ 0.2 pN on a typical 10 pN force value), whereas the relative error for distance values is < 0.1% (σd< 10 nm on a 10 µm distance value).

As explained in the main text, this implies that when performing a nonlinear least-squares fit, the force should be the dependent variable. For fitting the extensible worm-like chain, the eWLC equation [Eq. (3.2)] was therefore inverted using Mathematica (Wolfram, USA). The twistable worm-like chain [Eq. (3.3)] required a numerical inversion, since no closed-form solution could be found.

The importance of fitting the right representation of the WLC is highlighted in the main text, Fig. 3.4, which shows a strong dependence of the Lp fit results on the data range used for fitting. Fig. 3.6 shows the analogous figure for the stretch modulus S.

Global fitting

To improve the accuracy of the least-squares fits, and overcome overfitting issues due to the large number of free fit parameters, we implemented a global fitting approach. Force–extension curves for all molecules in an ensemble (i.e., all data associated with a particular magnesium concentration) were fit simultaneously, while sharing the physical parameters (i.e., Lp, S, g0, g1). This basically con-strained these physical parameters to have the same value for all fitted curves, while any other fit parameters were free to vary per-molecule. As an example: when fitting N force–extension curves with the eWLC, with the two offsets d0 and F0 as described below, one would fit all N datasets in one go, with a total of 2N + 2 fit parameters (Lp and S shared, and the two offsets per molecule). Nonlinear least-squares fitting further proceeded as usual.

Data alignment

As described in the main text, each recorded force–extension curve contains three systematic measurement errors:

S (nN) 0 1.25 2.5 3.75 5 d_min Force d_min / Lc 0.6 0.7 0.8 0.9

3.6 Methods

Alignment using global fit

Distance (μm) Force (pN) 0 5 10 15 20 25 30 12 13 14 15 16 17

Alignment using individual fits

(a) (b)

Figure 3.7 | Comparison of data alignment of force–extension curves. 10 mea-sured force–extension curves were aligned using (a) individual fits of each of the force–extension curves in an ensemble, and (b) a global fit. Only a global fit pro-duces accurate values for the offsets F0and d0.

To correct for the two offsets, we first fitted all data below the eWLC limit of 30 pN with an amended version of Eq. (3.2) that includes the offsets F0and d0:

d − d0 Lc = 1 − 1 2 kBT (F − F0)Lp !1/2 +(F − F0) S (3.5)

Since the contour length Lc was fixed to its theoretical value of 16.5 µm (for 48.5 kb λ DNA), this gave a total of 4 fit parameters during the global fit, 2 of which were shared (Lp, S), and 2 of which were non-shared (d0, F0). The values for the offsets found during this global fit were then subtracted from the data.

Note that using a global fit is of paramount importance, as illustrated in Fig. 3.7: only a global fit produces accurate values for the offsets F0and d0. This can be verified using a simulated dataset (Fig. 3.8): especially for the distance offset d0, the global fit performs significantly better. Performance of the global fit does depend on the measurement error δF . In the simulations of Fig. 3.8, global fitting only outperforms individual fitting as long as δF _{. 5%.}

Actual Found −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 F₀ (pN) Actual Found −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 d₀ (μm) (a) (b)

3.6 Methods

Figure 3.9 | No correlation between overstretching force and magnesium con-centration is observed. The force Fos at which the overstretching plateau occurs, is measured as the average force between d = 18.5 µm and d = 20 µm, and plotted versus magnesium concentration. Within the resolution of our instrument, there is no correlation between the two.

overlapping overstretching plateaus. The ‘target’ value of Fosis set to the average of the full dataset. Within the resolution of our instrument, there appears to be no correlation between magnesium concentration and overstretching plateau force (Fig. 3.9). Therefore, we rescale all data to the same overstretching force.

tWLC analysis

For the full tWLC analysis (Fig. 3.5), we first performed alignment of the full dataset, as discussed above. We then removed all data points above a maximum force Fmax, since the tWLC did not fit the full force–extension curves including overstretching plateau. We performed a ‘sweep’ of values for Fmax within each magnesium concentration ensemble. The Fmaxvalue that maximized, within each magnesium concentration ensemble, the coefficient of determination (R2_{) of the} fit, was taken as the final value (see Fig. 3.10). These final values are shown in Fig. 3.11.

0.9988 0.9990 0.9992 0.9994 0.9996 R2 40 50 60 70 15 20 25 30 g₁ (nm) 40 50 60 70 −1400 −1200 −1000 −800 −600 −400 g0 (pN nm) 40 50 60 70 1800 1750 1840 1900 S (pN) 40 50 60 70 F_max (pN) F_max (pN) F_max (pN) 42 43 42.5 L_p (nm) 40 50 60 70 F_max (pN) F_max (pN)

Figure 3.10 | tWLC fit results for different values of the maximum force Fmax. Data measured at zero magnesium concentration. The fit that maximizes, within this concentration ensemble, the coefficient of determination R2_{, is taken as the final} value (vertical dashed line). (Error bars: bootstrap error).

[Mg2+_{] (mM)} Fmax (pN) 62 63 64 65 0 50 100 150

3.6 Methods

eWLC verification

As a control, we also fitted the eWLC to the data using two different approaches: 1. We fitted each of the force–extension curves, prior to alignment, using the inverted version of Eq. (3.2) (eWLC). This resulted in, for each magnesium concentration, a distribution of Lpand S values across molecules. For each concentration, we took the mean and S.E.M. of that distribution.

2. We took the aligned data, and performed a global fit of the eWLC to the data for each magnesium concentration. Bootstrap errors were calculated for the error bars.

Note that in both approaches, only data below 30 pN, the upper limit of va-lidity of the eWLC, was used. Including data at higher forces leads to values of especially S that are too low (data not shown).

As shown in Fig. 3.12, both of these alternative approaches agreed well with the analysis as presented in Fig. 3.5. Note that the individual fits are less accurate, due to the aforementioned issues with obtaining distance and force offset values from individual fits.

Direct fitting of

|g(F )|

By rewriting Eq. 3.3, we find that we can express |g(F )|2 as a function of the data: |g(F )|2= SC − CF/   d Lc − 1 + 1 2 kBT F Lp !1/2  (3.6)

[Mg2+] (mM) S (nN) 1.6 1.7 1.8 1.9 0 50 100 150 [Mg2+] (mM) Lp (nm) 38 40 42 44 46 48 0 50 100 150 (a) (b)

Raw data, individual fits Fit combined aligned data tWLC analysis

Figure 3.12 | eWLC fits to the data, as a verification of the values for Lp and S in Fig. 3.5. Shown are (i) mean and SEM for individual fits to ‘raw’ (i.e., unaligned) data (blue, •); (ii) global fits of the eWLC to aligned data (red, ◦; error bars: bootstrap error); and (iii) values from the tWLC analysis in Fig. 3.5 (yellow −−). (a) Values for the persistence length Lp, and (b) stretch modulus S .

[Mg2+] (mM) 0 50 100 150 13 14 15 16 17 19 18 g1 (nm) [Mg2+] (mM) F (pN) 0 50 100 150 −750 −700 −650 −600 −550 −500 g0 (pN nm) 0 20 40 60 0 200 400 600 800 1000 |g (F )| (pN nm) (a) (b) (c)

3.7 References

Acknowledgments

We would like to thank T.T. Perkins for suggesting the initial experiment;

M.C.M. de Gunst for useful discussions; and the manuscript referees for

valuable suggestions. This work has been supported by grants from the

Foundation for Fundamental Research on Matter (FOM), which is part

of the Netherlands Organization for Scientific Research (NWO), and the

European Research Council (starting grant).

3.7 References

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3.7 References

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