Quantifying Molecular Forces with Serially Connected Force Sensors

Abstract

To understand the mechanical forces involved in cell adhesion, molecular force sensors have been developed to study tension through adhesion proteins. Recently, a class of molecular force sensors called tension gauge tethers (TGTs) have been developed that rely on irreversible force-dependent dissociation of a DNA duplex to study cell adhesion forces. Although the TGT offers a high signal-to-noise ratio and is ideal for studying fast/single-molecular adhesion processes, quantitative interpretation of experimental results has been challenging. Here, we use a computational approach to investigate how TGT fluorescence readout can be quantitatively interpreted. In particular, we studied force sensors made of a single TGT, multiplexed single TGTs, and two TGTs connected in series. Our results showed that fluorescence readout using a single TGT can result from drastically different combinations of force history and adhesion event density that span orders of magnitude. In addition, the apparent behavior of the TGT is influenced by the tethered receptor-ligand, making it necessary to calibrate the TGT with every new receptor-ligand. To solve this problem, we proposed a system of two serially connected TGTs. Our result shows that not only is the ratiometric readout of serial TGT independent of the choice of receptor-ligand, it is able to reconstruct force history with sub-pN force resolution. This is also not possible by simply multiplexing different types of TGTs together. Last, we systematically investigated how the sequence composition of the two serially connected TGTs can be tuned to achieve different dynamic range. This computational study demonstrated how serially connected irreversible molecular dissociation processes can accurately quantify molecular force and laid the foundation for subsequent experimental studies.

Introduction

Cell adhesion is a complex process involved in many biological systems, from cell proliferation and differentiation to cell-cell communication, organism development, and disease processes. At the molecular level, cell adhesion to its environment or other cells is achieved through adhesion bonds formed by specific receptor-ligand interactions. Mechanical forces across these adhesion bonds are dynamically regulated for cells to sense their environment and control their functions. Therefore, monitoring the dynamics of molecular force is crucial to understanding cell adhesion.

One approach to understanding the molecular adhesion bond is by single-molecule force spectroscopy, in which the interactions between a single pair of receptor and ligand are physically ruptured by force. This has been done through a number of high-resolution techniques (atomic force microscopy, optical tweezers, and magnetic tweezers) and massively parallel techniques (centrifugal force microscopy and acoustic force spectroscopy) (1, 2, 3).When a single-adhesion bond is subjected to a constant force or an increasing force at a constant force loading rate, its behavior is characterized by either its lifetime (τ) or the mean rupture force (F_r). Because bond rupturing is an intrinsically nonequilibrium process, τ or F_r measurements at a single force or loading rate are insufficient to describe the full behavior of an adhesion bond. In addition, without measuring molecular adhesion forces in living systems, it is impossible to determine the biologically relevant range of force or loading rate for the adhesion bond of interest.

The development of molecular force sensors aims to address this question. There are two major classes of force sensors using either reversible or irreversible molecular force sensing elements. Reversible force sensors can be further divided into two subcategories: analog and digital. The analog design utilizes the elastomeric response of individual polymers or proteins to measure the instantaneous forces on receptors (4, 5, 6, 7, 8). At the single-molecule level, they are excellent force reporters (Fig. 1 a). However, in live cell situations in which individual sensors cannot be spatially resolved, the analog nature of the sensors also makes it difficult to tease out the heterogeneity of molecular force and adhesion bond density. In addition, the tunability of their dynamic range is limited with only a handful of proteins and polymers (4, 5, 6). Reversible digital force sensors utilize the reversible two-state structural transitions of biomolecules, such as DNA hairpin (9, 10, 11) and FL peptide (12). In these designs, the folded domain undergoes a rapid structural transition between the folded and unfolded state to reach a dynamic equilibrium (Fig. 1 b). As force increases, the folding rate decreases, whereas the unfolding rate increases, resulting in the biomolecule spending more time in the unfolded state.

Figure 1.

Illustration of three major classes of molecular force sensors and their fluorescence signal interpretation. (a) Reversible analog force sensor with an elastomer as the force-sensing element is shown (top). Elastomer end-to-end distance increases in response to tension (middle). A pair of fluorophores can produce a fluorescence resonance energy transfer signal that directly correlates to the applied force (bottom). (b) A reversible digital force sensor utilizing the force-dependent two-state structural transition (such as unfolding) of a biomolecule as the force sensing element is shown (top). The force-sensing element is best described by an unfolding probability as a function of force (middle) and gives two distinct fluorescence signal levels at the single-molecule level. The dashed lines indicate the time-averaged signal level as a function of force (bottom). (c) Irreversible force sensors rely on the rupture of a tethered adhesion bond to inform about the relative strength of the adhesion bond of interest (top). The response of the sensor is characterized by its force-dependent dissociation lifetime (middle), which can be measured from the time-dependent population statistics of the rupture events (bottom). To see this figure in color, go online.

The time-averaged fluorescence signal from the system is essentially digital around a single threshold force if the transition is sharp. The advantage of such design is that the fluorescence signal reports the number of sensors above the force threshold without having to spatially resolve individual sensors. However, the drawback is that forces are reported in a binary fashion (“on” or “off”) around a single threshold. This trade-off of force resolution for sensor density information makes it difficult to probe the magnitude of molecular force even with multiplexing such sensors (9, 12, 13). Nonreversible force sensors, such as the tension gauge tether (TGT) and the nano-yoyo, utilize the force-dependent rupture of a calibrated bond as the sensor to determine the relative strength of the target receptor-ligand interaction (14, 15, 16, 17, 18). Fluorescence signal is “on” when the sensor is ruptured or “off” if it is intact (Fig. 1 c). The digital behavior allows one to determine the density of sensors in the ruptured state via fluorescence and create adhesion footprints that map exactly where the adhesion events happened (16). This technique provides excellent fluorescence signal-to-noise ratio because the background signal from intact sensors can be essentially suppressed. However, quantitative interpretation of molecular force using irreversible force sensors has been challenging. In particular, Mosayebi and co-workers have pointed out that “tension tolerance,” defined as a single critical force, is a function of the timescale of the observation and, hence, an inappropriate parameter to quantify the nonequilibrium adhesion process (13).

In this report, using simulation with TGT as an example, we first illustrate the limitations of irreversible force sensors—how the same fluorescence readout can result from drastically different mechanical histories, and how the target receptor modulates TGT characterization. We then developed a strategy using two serially connected TGT as internal calibration pairs to bypass these limitations to quantitatively extract force history. Last, we described how to design the serial TGT system to be able to achieve a desirable force dynamic range.

Methods

Monte Carlo simulations

The Monte Carlo method was used to simulate the stochastic bond breaking of components (TGT and RGD-integrin α₅β₁) in the TGT-receptor complex. The simulation breaks time into short time steps (Δt) and assumes that force is constant within each time step, whereas force can be arbitrarily changed between force steps to achieve any desirable force history. Because the survival probability P of a bond is defined by:

$$P\left(t,F\right)=\exp \left(-t/\tau \left(F\right)\right),$$ (1)

where t is time and τ(F) is the force-dependent lifetime of the bond. Probability within each simulation time step is therefore P(Δt, F) and is compared to a random number between 0 and 1 to determine whether the bond survives within the time step. The simulation stops when either one of the components (TGT or RGD-integrin) breaks. The simulation code was written in MATLAB (The MathWorks, Natick, MA) (R2017a) and was run on desktop grade central processing unit (Intel i7-6820HK) and graphics processing unit (GPU) (Nvidia GTX 980M with 1536 CUDA cores).

Free-energy landscape parameters and bond lifetime calculations

We developed a free-energy landscape model similar to the ones by Woodside et al. and Mosayebi et al. (13, 19), in which both the hybridization free energy and the single-stranded DNA (ssDNA) elastic-free energy along the unzipping (end-to-end distance, x) reaction coordinate are incorporated. At any given x, there is an ensemble of conformations with different numbers of basepairs remaining in the DNA structure. We first calculate the partition function at each x by taking into account all possible conformations at x and their associated free energy contribution from secondary structure, ssDNA stretching, and external force. The expected free energy along x is used to construct the free-energy landscape. The transition state is subsequently defined as the local maximum of the free-energy landscape. To generalize the free-energy landscape parameter for a particular sequence length and CG content, we took 100 random sequences at each length/CG content combination to produce their average free-energy barrier parameters (see Fig. S4; Table S1). The average unzipping free-energy barrier height and width were used for subsequent calculations. The free-energy landscapes parameters calculated this way fit closely to the experimental DNA unzipping experimental data from Woodside et al. (19).

In the context of this work, we consider the unzipping of TGT as a single “bond rupture” event, in which the term bond refer to the DNA duplex instead of individual basepairs or H-bonds. The bond lifetime for TGT unzipping is calculated by:

$$\tau \left(F\right)={\tau }^{\ast }\exp \left(\frac{\mathit{\Delta }{G}_0^{‡}+\mathit{\Delta }{G}_{stretch}-F{x}^{‡}}{k_{B}T}\right),$$ (2)

where τ^∗ = 4.12 × 10⁻⁵ s is the intrinsic time constant for barrier crossing calculated using least squares fit of our model to the experimental results of Woodside et al. (19). k_B is the Boltzmann constant; T is the temperature; F is the force applied to the bond; ΔG^‡₀ is the height of the free-energy barrier at zero force calculated by Mfold (20); x^‡ is the barrier width, defined as the distance from the folded state to the transition state along the reaction coordinate of end-to-end distance; and ΔG_stretch is the elastic contribution to the free energy from ssDNA stretching according to Woodside et al. (19):

$$\mathit{\Delta }{G}_{stretch}\left(x^{‡}\right)=\frac{k_{B}T{L}_0}{4L_p\left(1-x^{‡}/L_0\right)}\left[3{\left(x^{‡}/L_0\right)}^2-2{\left(x^{‡}/L_0\right)}^3\right],$$ (3)

where L_p is the persistence length of ssDNA at 1 nm/nucleotide, L₀ is the contour length of freed ssDNA after unzipping with 0.59 nm/nucleotide, and F_max is the force causing the flattening of the free-energy landscape defined as (ΔG^‡₀+ΔG_stretch)/x^‡. Considering N bonds are serially connected and assume no two bonds in the series rupture simultaneously, the probability that, given a bond rupture event, it was the i^th bond that ruptured at a given force F is

$$P_i\left(F\right)=\frac{{\tau }_i^{-1}\left(F\right)}{\sum _j{\tau }_j^{-1}\left(F\right)}.$$ (4)

Simulation parameters

All simulations were run at T = 298 K. Constant force simulations were carried at 0.1 pN increments starting from 0 pN to F_max for double-stranded DNA. The temporal resolution (Δt) was independently determined for each force as a function of τ(F) as being equal to the 1/100 of the smallest τ(F). Simulations were run for 10⁴ trials per simulation condition. Both types of simulation were run on the GPU when possible. All the parameters above apply unless otherwise noted.

Results

Monte Carlo simulation of force-dependent bond rupture kinetics

To simulate bond rupture behavior, we first developed a Monte Carlo simulation to capture the stochastic rupture kinetics and statistics of a system of bonds with arbitrary force history and temporal resolution. With GPU computing and scalable time steps, we are able to probe the force-dependent dissociation behavior of 10⁵ bonds over 12 orders of magnitude in time on a single personal computer. Experimentally determined force-dependent dissociation parameters for DNA unzipping (19) and α₅β₁ integrin-RGD (FNIII_7–10) interaction (21) were used in this study. We used the Bell-Evans free-energy landscape model with ssDNA elasticity correction to model DNA unzipping (19). To simulate the force-dependent dissociation of different DNA sequences, the dependencies of the unfolding free-energy barrier height (ΔG^‡ = ΔG^‡₀ + ΔG_stretch) and width (x^‡) on the length and CG content were extrapolated (Fig. 2, a and b). Our simulation produces excellent agreement with analytical and numerical solutions under both constant force and constant loading rate conditions (Fig. 2 c, see Supporting Materials and Methods for details). With this simulation tool, we are able to obtain bond rupture kinetics and statistics at a high temporal resolution for systems of adhesion bonds subjected to an arbitrary mechanical history (Fig. 2, d and e, see Supporting Materials and Methods for details).

Figure 2.

Monte Carlo simulation and validation of DNA unzipping kinetics. (a) Shown is a schematic of DNA unzipping, one of the core force sensing elements in TGT. (b) Shown is a schematic of the force effect on the free-energy landscape of DNA unzipping. (c) The DNA unzipping free-energy barrier height (ΔG^‡) and width (x^‡) as a function of DNA length (bp) and number of CG (bp) are shown. (d) Survival probability of intact DNA (L16/CG6) as a function of time over a range of constant forces (1–12 pN) is shown. (e) Probability density of DNA (L16/CG6) rupture force over a range of force loading rate (0.1–10 pN/s) is shown. Circles indicate the results from simulation; solid lines indicate numerical solutions. To see this figure in color, go online.

TGT signal is modulated by receptor-ligand bond and is nonunique

The force-fluorescence response of the sensing element in a molecular force sensor is usually characterized separately by single-molecule force spectroscopy. It is assumed that the force-fluorescence response remains identical when the sensing element is linked to the target receptor. This is a valid assumption under biologically relevant timescales for reversible force sensors because the sensing element is in a conformational dynamic equilibrium and is able to report instantaneous force via fluorescence. However, a quantitative interpretation of the fluorescence signal generated from an irreversible force sensor is challenging because it is a cumulative signal that depends on several factors that cannot be easily teased apart. To illustrate this, we simulated the time- and force-dependent unzipping of DNA (L20/CG10), both in the presence and absence of conjugated RGD peptides bound to α₅β₁ integrins. The unzipping of DNAs can be experimentally observed via fluorescence gains, whereas the rupture of the receptor-ligand bond keeps the DNA intact, generating no fluorescence signals (Fig. 3, a and b (16)).

Figure 3.

Simulation of DNA unzipping in the absence and presence of a receptor-ligand pair. (a) Fluorescence signal is generated whenever the top strand of the DNA is unzipped. (b) In the presence of a receptor-ligand pair, the fluorescence signal is generated only in the outcome in which DNA is unzipped. (c) Observed fluorescence curves as a function of time at different forces (8–13 pN) and surface density (vertical planes, 10³–10⁵ events/unit area) are shown. The intersection (dot) of each curve with the horizontal plane indicates a unique combination of force, surface density, and time duration that result in the same fluorescence readout. (d and e) The fraction of DNA rupture events as a function of time over a range of constant forces corresponding to a scenario in (a) and (b), respectively, are shown. (f and g) The probability density function of DNA rupture force over a range of loading rates, corresponding to scenarios in (a) and (b), respectively, is shown. (h) The apparent half-life and fraction of DNA rupture as a function of force is calculated from (d to e). (i) The mean DNA rupture force and fraction as a function of loading rate is calculated from (f to g). All DNA in the simulations are L20/CG10. To see this figure in color, go online.

A constant force acting directly to unzip DNA (Fig. 3 a) will eventually unzip all the DNA; as shown in Fig. 3 d, DNA rupture fraction converges to 1 for all forces. This is because force-dependent dissociation is irreversible and happens on the first pass across the free-energy barrier. The barrier-crossing rate increases with increasing force and is well described by previous models and experimental evidence (19). In this idealized system, if the surface density of TGT and the duration of force are known, the magnitude of force can theoretically be calculated.

In the case in which forces are applied to stretch DNA together with the receptor/ligand pair (Fig. 3 b), the apparent DNA unzipping kinetics are modulated by the force-dependent rupture of the receptor-ligand bond as seen in Fig. 3 e. At higher forces (14 pN), DNA unzipping kinetics curve resembles that of DNA unzipping alone (Fig. 3 d); this is because the DNA lifetime at 14 pN is significantly shorter than that of the receptor-ligand bond (Fig. S1). As force decreases, the dissociation rate of DNA decreases and eventually drops below the dissociation rate of the receptor-ligand bond. Hence, the receptor-ligand bond rupture events contribute to the majority of all the rupture events, whereas DNA unzipping makes up a small fraction (Fig. 3 e). Because of this, different force histories can produce the same level of fluorescence intensity during an imaging experiment. Therefore, quantitative interpretation of molecular forces based on fluorescence intensity alone is extremely challenging (Fig. 3 c). In addition, the apparent kinetic half-life of DNA unzipping is significantly shortened in the presence of the receptor-ligand bond (Fig. 3 h). In an experiment where the only observable is fluorescence intensity change over time, the apparent lower DNA half-life would lead to an over-estimation of force. Although the physical behavior of the DNA unzipping is ideally independent of any attached receptor-ligand, the deviation is the result of a receptor-ligand breaking event exhausting the total number of tethers under force. The effect is especially pronounced at lower molecular forces. Similarly, in constant loading rate scenarios, the apparent rupture force distribution in the presence of receptor-ligand is significantly skewed and shifted toward lower forces (Fig. 3, f and g). As force ramps up, an increasing number of receptor-ligand bonds break at lower forces, whereas the number of remaining DNA tethers available at higher forces decrease, causing a systematic suppression of DNA unzipping events at high force. In a constant loading rate experiment, this would lead to an overestimation of the rupture force; the effect is increasingly prominent at biologically relevant loading rates (22) below <10 pN/s (Fig. 3 i). Therefore, irreversible force sensors characterized in isolation can only be used to model fluorescence intensities of TGTs in experiments at high force and high loading rate regimes, in which the force-dependent dissociation rate of the sensor is significantly higher than that of the receptor-ligand pair.

Under both constant force and constant loading rate conditions, the fluorescence intensity resulting from DNA rupture is a function of force magnitude, duration, surface density, and the rupture characteristics of the receptor-ligand bond. Hence, a single snapshot of fluorescence image in cell experiments is insufficient to quantify force history. In addition, quantitative assessment of TGT response in a real experimental system would require the characterization of the force-dependent dissociation of the receptor-ligand bond through dynamic single-molecule force spectroscopy. This inevitably makes it difficult to use TGT as a modular component in various systems to study molecular adhesion.

Force quantification via two force sensors in series

To address the challenges above, we investigated systems composed of two TGT elements in a series (Fig. 4 a) and in parallel (Fig. 4 b). In the serial system, two different DNA sensor elements (DNA₁, L16/CG6 and DNA₂, L17/CG6) are serially connected to the receptor-ligand bond. When subjected to force, there are three possible outcomes (at t = ∞): 1) receptor-ligand bond ruptures, 2) DNA₁ ruptures, or 3) DNA₂ ruptures. The rupture events of DNA₁ and DNA₂ can be detected via fluorescently labeled (red and green) complementary ssDNA, where R and G are fractions of all the rupture events. We use three systems under constant force (range in 0–15 pN): 1) without conjugate, 2) conjugated to RGD-α₅β₁ integrin, and 3) conjugated to DNA3 (L18/CG6 in the unzipping geometry). Whereas the profiles of R and G are distinct across the three conditions (Fig. 4 c), all three showed an identical G/R ratio, which is a one-to-one function of force (Fig. 4 e). When DNA₁ and DNA₂ are placed in a series, the pair form an internal cross-reference that is undisturbed by the force-dependent rupture behavior of the receptor-ligand bond. Therefore, the magnitude of force determined from G/R is completely independent of the receptor-ligand bond it probes, in contrast to the case of a single DNA. This makes it possible to apply serial DNA sensors calibrated by single-molecule force spectroscopy to any adhesion bonds, knowing the calibration results will hold. Furthermore, the dimensionless G/R makes force quantification independent of surface density, which is another factor that readouts from individual TGTs cannot distinguish.

Figure 4.

Force- and time-dependent signals from serial and parallel TGTs. (a and b) Schematics of the serial and parallel configurations are shown. The red and green DNAs (L16/CG6 and L17/CG6) together make up the sensing element and will produce a red (R) and green (G) fluorescence signal when ruptured. The gray DNA (L18/CG6) and integrin-RGD are two distinct target adhesion bonds tested in this simulation. (c and d) Shown is the rupture fraction of each component as a function of force at t = ∞; colors of lines match the components in (a) and (b). (e and f) G/R as a function of force with line style (solid, dashed, and dotted) indicate the three different cases, matching (c) and (d). (g and h) Shown is the rupture fraction of each component (stacked) as a function of time at a constant force of 8 pN; colors match the components in (a) and (b). (i and j) G/R as a function of force with line style (solid, dashed, and dotted) indicate the three different cases, matching (g) and (h). Initial fluctuation in G/R is due to stochastic sampling of a small sample size. To see this figure in color, go online.

Next, we studied the time dependence of G/R under constant force conditions. The cumulative bond rupture events over time of each component are shown in Fig. 4 g. Even though, individually, R and G are both strongly dependent on the connected receptor-ligand pair, G/R is time invariant at any given force (Fig. 4 i). This implies that no matter when an observation is made or how long the constant force duration is, a single fluorescence snapshot will allow for force quantification with the only assumption being a constant force scenario. Therefore, G/R is a monotonic function of force and force alone under a constant force scenario (see Supporting Materials and Methods for details). Because of this, if one assumes force is relatively constant in short time intervals, ΔG/ΔR can be used to report the magnitude of force in the time interval, where ΔG and ΔR are incremental changes in fluorescence intensity. This would allow for force determination over time.

Last, we investigated the same pair of DNA sensors in a parallel configuration (Fig. 4 b). In this configuration, DNA₁ and DNA₂ are independently connected to the target receptor, which is often how multiplexing experiments were designed. We performed the same set of characterizations (Fig. 4, d, f, h, and j) as we did for the serial configuration (Fig. 4, c, e, g, and i). Our simulation shows that the force-dependent profile of G/R is dependent on the conjugated receptor-ligand pair (Fig. 4 f). Hence, this configuration suffers the same issues as the single DNA sensor system in which the G/R is modulated by the targeted adhesion molecule modulation. Simulation of the time-dependent behavior of the system (Fig. 4, h and j) shows that G/R changes over time in a conjugate-dependent manner. Therefore, the ratio of incremental fluorescence change ΔG/ΔR is also time and conjugate dependent. This does not come as a surprise as DNA₁ and DNA₂ rupture independently. There is no function f(G,R,F) in the parallel configuration that can make the result converge into one function as in the case of the serial configuration.

From the above results, two serially connected DNAs can be treated as a single force sensing element in which force magnitude at a given time interval can be reported through the ΔG/ΔR ratio. The serial sensor design, for the first time, offers the possibility to use irreversible sensors to measure the magnitude of molecular force, which is not possible using a single irreversible sensor or through parallel multiplexing of sensors.

Probing force history and surface receptor density

Because the ratio of fluorescence signal change ΔG/ΔR is only dependent on force, a force history can be reconstructed from monitoring fluorescence changes over time. We simulated constant loading rate (1 pN/s) and oscillatory force (8.75 and 7.75 pN intervals) scenarios to demonstrate how sub-pN force resolution can be achieved using serial TGT designs. A total of 10⁵ serial TGT (L16/CG6 and L17/CG6) with RGD-α₅β₁ integrin were simulated in each scenario. Fluorescence signals R and G are recorded to mimic experimental observables, which are converted to ΔG/ΔR using 50-ms time intervals to reconstruct forces using the established calibration curve (Fig. 4 e) for this TGT pair. The results show that force histories reconstructed from ΔG/ΔR closely reproduce the actual force history (Fig. 5, a and b) and that sub-pN changes are clearly detectable. A step-by-step force history reconstruction from G and R signal is shown in Fig. 5, a and b.

Figure 5.

Force history and surface density reconstruction. (a) Shown is the reconstruction of force history from a constant loading rate simulation with the applied force starting at 7.5 pN and increasing at 1 pN/s. (b) Shown is the reconstruction of force history under an oscillatory force regime in which the force alternated between 8.75 and 7.75 pN. In both (a) and (b), top panels are the total count of R (red) and G (green) signal as observed experimentally. Middle panels are the changes ΔR (red) and ΔG (green) of R and G signals within each time interval Δt = 50 ms as well as their ratio ΔG/ΔR (blue). Bottom panels are forces reconstructed from ΔG/ΔR using Fig. 4e. (c) Force as a function of R + G at different surface adhesion event densities (100, 200, 500, 1000, 2000, and 5000 per diffraction limited spot) is shown. The solid curves are the ground truth levels of force and the corresponding R + G at a given surface density. Circles are the average values from 100 constant force simulations. The error bars are SD of force (vertical bars) and R + G (horizontal bars) from the simulations. The uncertainty increases as the total number of rupture events R + G decreases (see Fig. S3 for more details). To see this figure in color, go online.

Next, we demonstrate that the receptor surface density can be calculated even though fluorescence signals from TGT rupture only report a fraction of adhesion events. One of the challenges in interpreting fluorescence signals from molecular force sensor is to tease apart contributions due to force and adhesion event density. This is particularly difficult in the case of a single TGT, as illustrated in Fig. 3 c; depending on the force history, only a fraction of TGT can rupture to produce a fluorescence signal within a diffraction limited area. Serial TGT systems can bypass this issue, as there is a singular solution that relates 1) the magnitude of force, 2) the fluorescence intensity, and 3) the number of actual adhesion events within the same diffraction limited area that gives rise to the force and total fluorescence intensity (Fig. 5 c). As shown in Fig. 5 c, each solid curve represents the force versus G + R at a given adhesion event surface density (100, 200, 500, 1000, and 2000 per diffraction limited area). Because these curves do not intersect, given a pair of G and R values, both the force (through G/R) and the actual surface adhesion event density (through the curve it falls on) can be determined (Fig. 5 c). This of course is not without its limitation; statistical variations of G and R due to stochastic bond rupture give rise to uncertainties in force and surface density, as illustrated in Fig. 5 c. Low surface density combining with low force will result in low G + R, hence greater uncertainty in force, as shown in data points with error bars in Fig. 5 c.

Rational design of serial force probes

There is a large DNA sequence space to explore for serial TGT designs. Because the behavior of force-dependent DNA unzipping can be well characterized by the length and CG content of DNA, these are the only two parameters used to develop a serial TGT design strategy. We systematically evaluated every possible pair of DNA with length ranging from 15 to 30 bp and CG content from 0 to 100%.

First, we looked at how length and CG content affect the force-dependent lifetime τ of a single DNA before the irreversible unzipping occurs (Fig. S2, a–c). The slope of log(τ) versus F plot is dictated by the unzipping free-energy barrier width, whereas a vertical shift is dictated by the ratio of barrier height and width (Fig. 2 b). The free-energy barrier width is primarily controlled by the length of the DNA, whereas the barrier height is a function of both length and sequence. Hence, increasing CG content while keeping the DNA length constant shifts log(τ) higher, whereas changing length controls the tilt of log(τ). Fig. S2, d–f compares the log(τ) versus F curve of L20/CG10 DNA against all DNAs with length range 10–30 bp and CG content 0–100%. As expected, a similar length and CG content brings the log(τ) versus F curve closer to that of L20/CG10 DNA as measured by the mean-square distance between the curves (Fig. S2, e and f). For each pair of DNA with RGD-integrin, we defined a force quantification range (FQR) (FQR = F_max − F_min) where force quantification is feasible through fluorescence ratio (ΔG/ΔR). There are three factors that restrict the FQR for each DNA pair: 1) insufficient fluorescence signal due to RGD-integrin bond breaking. At low forces, RGD-integrin bond dissociation dominates all the dissociation events, yielding a low fluorescence signal that sets the lower limit of the force detection range. We defined the lower limit as the force in which DNA unzipping makes up less than 10% of all the dissociation events (Fig. 6 b). 2) The force sensitivity of the DNA pair. If the change of ΔG/ΔR as a function of force is insignificant, force cannot be accurately reconstructed. In addition, the fluorescence signal from both DNA must be detectable. We defined the useful range for a DNA pair because the rupture fraction (f = R/(R + G)) of either DNA₁ (R) or DNA₂ (G) must change at least 1% per 1 pN change in force (i.e., df/dF > 0.01 pN⁻¹) (Fig. 6 c). 3) The upper limit of unzipping rate that can be predicted using energy landscape theory. This defines the upper limit in which the force tilts the free-energy landscape far enough that there is no longer a free-energy barrier between the folded and unfolded structure. The DNA unzipping lifetime cannot be predicted above this point and we decided to use this as the upper limit of FQR (Fig. 6 b). Combining the above three constraints, the overall F_min and F_max defines the FQR (Fig. 6 c).

Figure 6.

Systematic characterization of DNA pairs for serial TGTs. (a) Shown is the force-dependent dissociation lifetime of L16/CG6 (red), L17/CG6 (green), and RGD-integrin (gray). (b) Shown is the rupture fraction of each component (L16/CG6, L17/CG6, and RGD-integrin) in a serial TGT system at different forces. Gray dashed lines indicate the constraints from 10% DNA rupture events (left) and free-energy landscape prediction limit (right). (c) Shown is the fraction of DNA1 rupture among all DNA ruptures (f = R/(R + G)) and force sensitivity (df/dF) of serial TGT system as a function of force. The horizontal dashed line indicates the force sensitivity threshold for df/dF > 0.01/pN. The overall F_min and F_max are defined by the region of df/dF within the three constraints marked by the dashed lines. (d) Shown is a scatter plot of F_max versus F_min of serial TGT for every pair of DNA with length between 15 and 30 bp. The color map indicates the length difference (left) and number of CG differences (right) of the DNA pair. (e) Three-dimensional scatter plot shows how F_mid and FQR depend on DNA₂ sequence composition for two DNA₁ examples: L20/CG20 and L16/CG6. To see this figure in color, go online.

To understand what factors control F_min and F_max, we looked at the differences in length and in number of CG bases between the two serially connected DNAs (Fig. 6 d). Each dot in Fig. 6 d represents one pair of DNAs, with its location indicating its F_min and F_max. The large spread shows the large tunability of the serial TGT system—there are pairs that can quantify in the 2–10 pN range on the low end and 15–20 pN range on the high end. The DNA pair with larger FQR tend to be the ones with a smaller length and CG differences (Fig. 6 d). This is because a few basepairs of difference in length and CG is generally sufficient to produce ΔG and ΔR that respond distinctively at different forces, yet the sequence difference is not so large to ensure that both ΔG and ΔR are experimentally observable over a large force range. For illustration purposes, we picked two sequences for DNA₁ (L20/CG20 and L16/CG6) and systematically looked at how DNA₂ sequences affect the FQR. In Fig. 6 e, we plotted the force at midrange F_mid = (F_min + F_max)/2 as a function of the second pair’s length and number of CG and color-coded each dot by its FQR. Increasing CG while keeping length constant generally increases F_mid (Fig. 6 e), which is contributed to the up shift of the log(τ) curve (Fig. S2 c) of DNA₂. On the other hand, keeping the number of CGs in DNA₂ constant while changing its length does little to F_mid but affects the FQR significantly (Fig. 6 e). As the sequence composition of DNA₂ approaches that of DNA₁, FQR increases sharply. For L16/CG6, the sequence of DNA₂ that gives rise to the highest FQR is F17/CG7 and L16/CG16 for L20/CG20.

Discussion

Interpretation of fluorescence readout from irreversible force sensors such as the TGT requires great care. A single threshold force is likely insufficient to describe the force-dependent behavior of molecular dissociation. For instance, in the unzipping geometry, L15/CG15 is “weaker” than L30/CG15 below ∼12 pN, as shown by its lower unzipping lifetime τ (Fig. S2 a). However, at forces above ∼12 pN, L15/CG15 becomes “stronger” than L30/CG15. The crossover point at ∼12 pN for these two DNAs defines the force at which the two are equally strong with equal dissociation probability. Hence, the relative strength of different TGTs cannot be compared simply by a single threshold force assigned for each. Similarly, when a single TGT is tethered to a receptor-ligand bond, the crossover force is that which the lifetime of the TGT and receptor-ligand equals may serve as a better definition of the characteristic force that is specific to the TGT/receptor-ligand pair. Furthermore, we showed here that the apparent behavior of TGT is modulated by how receptor-ligand bonds respond to force. These make it difficult to 1) quantitatively interpret fluorescence images from TGT and 2) to quantitatively compare forces using the same TGT across different receptors. By placing two distinct TGTs in a series, they experience the same force and form an internally calibrated pair. Relative to each other, their rupture probability ratio is directly related to the force and is entirely independent of the receptor-ligand bond.

The serial sensor design scheme allows us to detect molecular forces using population statistics rather than individual sensors (Fig. 1). To our knowledge, this is the first time a design involving irreversible processes is able to provide an analog force readout, which has tremendous advantages in studying molecular forces. Because of the vast candidate choices and design possibilities utilizing molecular dissociations, there is a significant advantage in using serial sensor designs to achieve drastically different sensor dynamic ranges. It is important to keep in mind that force interpretation through ratiometric fluorescence detection in serial TGT assumes a constant force acting within the area of interest, which can be a diffraction-limited spot in conventional fluorescence microscopy or a smaller area in super-resolution microscopy. Although our method is unable to report force variations within a resolution-limited area, the force interpreted from such ratiometric measurement is closely related (but not exactly equal) to the average force within the area. This is largely because of the fact that the fluorescence ratio (G/R) is a monotonic function of force (Fig. 4 e). We expect super-resolution techniques such as stimulated emission depletion microscopy and DNA-PAINT will fully utilize the potential of serial TGT force sensors.

How does the serial TGT design compare to reversible force sensors? One of the challenges of the reversible sensor when used alone is the inability to tease apart the surface density of force events and the magnitude of force, even with the assumption of homogeneous force per unit area (e.g., within a diffraction limited pixel). For instance, 10% of sensors subjected to 10 pN of force may produce the same fluorescence readout as 20% of sensors subjected to 5 pN of force. This can be tackled (23) using a design in which one can detect simultaneously the force and know how many sensors are being pulled. By having two distinct sensors in one construct—one that responses to low force, hence any force application shows that it has been pulled and gives information about the surface density, whereas the other sensor provides more quantitative force information. This type of design usually utilizes quencher and fluorophore systems and may suffer from sensitivity issues, especially if the force-event surface density is low. In addition, such a system may not be the most suitable for fast dynamics cellular events, such as those in rolling adhesion in which the receptor is relatively sparse and the force application is brief (10–100 s of ms (16, 24)). A very high sensitivity yet quantitative solution is required. The TGT system is most suitable for such single-adhesion, high-speed adhesion events as illustrated in our previous work (16). Using this design, we hope to resolve this issue and provide quantitative information about the mean molecular force required for rolling adhesion. The application of the serial TGT is hence most suitable for such a system.

Conclusions

The involvement of mechanical processes in a wide array of biological processes makes their study and quantification increasingly important in the life sciences. Many methods have been developed and tested in the literature, each with their advantages and drawbacks. Here, we have introduced a new design to surmount some of the common issues faced when using molecular force sensors and validated it using Monte Carlo simulations. We have shown it to be independent of the adhesion molecule being studied as well as the surface receptor density. This design allows for the force, as well as the force history, to be determined for a wide range of biologically relevant adhesion molecules.

Author Contributions

Y.M. and I.T.S.L. designed the research. Y.M. carried out the simulation. Y.M. and I.T.S.L. analyzed the data. Y.M. and I.T.S.L. drafted the manuscript.

Editor: Alexander Dunn.