Computed Three-Dimensional Atomic Force Microscopy Images of Biopolymers Using the Jarzynski Equality.

Three-dimensional atomic force microscopy (3D-AFM) has resolved three-dimensional distributions of solvent molecules at solid-liquid interfaces at the subnanometer scale. This method is now being extended to the imaging of biopolymer assemblies such as chromosomes or proteins in cells, with the expectation of being able to resolve their three-dimensional structures. Here, we have developed a computational method to simulate 3D-AFM images of biopolymers by using the Jarzynski equality. It is found that some parts of the fiber structure of biopolymers are indeed resolved in the 3D-AFM image. The dependency of 3D-AFM images on the vertical scanning velocity is investigated, and optimum scanning velocities are found. It is also clarified that forces in nonequilibrium processes are measured in 3D-AFM measurements when the dynamics of polymers are slower than the scanning of the probe.

Three-dimensional atomic force microscopy (3D-AFM) is a cutting-edge technology to observe the distributions of solvent molecules at solid–liquid interfaces by mapping forces in 3D space.[1,2] It has been applied to mica,[1,3] lipid membranes,[4] proteins,[5] and DNA surfaces[6] to reveal their interfacial hydration structures. These observations of the average behavior of fluctuating water molecules suggest its capability of observing similar effects in mobile molecular chains in liquids, and there are already a few examples.[4,7,8] A recent paper reported that even the 3D structure of nanostructures in cells has been visualized with specially fabricated thin and long probes.[9] In principle, 3D-AFM technology offers the possibility to resolve the 3D organization of on-surface biopolymers such as chromosomes, cytoskeletons, and so on in cells, whose structures at the nanometer scale are still poorly known. To investigate such possibilities, there is an urgent demand to establish models computing 3D-AFM images of fluctuating biopolymers that can be linked to experiments.

Simulations of 3D-AFM images and a comparison of them with the experiments are essential to provide a theoretical basis for explaining what is resolved in imaging.[10] Accordingly, the simulation techniques to compute 3D-AFM images of solvents on surfaces were developed.[11] The free energy profile was first computed, and then, its derivative gave the so-called mean force to be compared with the measured forces in 3D-AFM experiments. To compute the free energy profile, umbrella sampling[11] and perturbation methods[11,12] were commonly employed. This strategy assumes that the system is in equilibrium throughout a cantilever oscillation cycle, which is only valid for the case where the motions of solvents are much faster than that of the scanning probe. Thus, an alternate method to simulate 3D-AFM images had to be developed for biopolymers since their motions are slower than the scanning velocity.[13]

In our previous paper,[13] we developed a method to visualize the surface topography of biopolymers by calculating the average forces during scanning, but this did not offer any insight into the interactions within the biopolymer necessary to understand 3D-AFM images, which also provide structural information inside the biopolymer. Here, we have developed a method to compute 3D-AFM images of biopolymers by using the Jarzynski equality that relates the work in the nonequilibrium process to the free energy difference.[14] It was recently shown that the Jarzynski equality is capable of estimating free energy even for glassy systems that are not in equilibrium, for which the thermodynamics integration is not applicable.[15] The Jarzynski equality has previously been used to calculate force–distance curves,[16−18] which were compared with the relevant AFM measurements.[19,20] This strategy is extended to map forces in the 3D space even inside biopolymers that are fluctuating in the liquid environments (Figure ), which enables us to see the 3D fiber structures of the polymer. A biopolymer and a probe were simulated using the bead–spring model.[13] The globular structure mimics chromosomes in the interphase (Figure a).[21,22] A thin and long probe was used as a model to mimic that used in a recent work;[9] this is long enough to penetrate inside the biopolymer (Figure b). The force–distance curves were computed at each x and y position using the Jarzynski equality, and then then all the curves were merged to give a 3D-AFM image (see details in the Supporting Information).

Figure 1

(a) Schematic of the simulation system and AFM probe model. There is one biopolymer composed of 2000 beads (mimicking a chromosome in the interphase) and one AFM probe composed of 50 beads in the system (only part of the probe is shown for clarity). The biopolymer is colored from red (one end) to blue (the other end). (b) A cross section of the biopolymer to illustrate the probe penetrating the polymer (see Movie S1).

Figure a shows the simulation system, and Figure b indicates an xz-slice of a 3D-AFM image at the center of polymer. Figure c is the color key for the image. The vertical scanning or penetration velocity (vscan) is 1 μm/s in Figure . The forces between the penetrating probe and the polymer are weakly attractive (approximately −20 pN) when the probe is outside the polymer, originating from the Lennard-Jones force between the probe and polymer in long-range. On the other hand, the forces when the probe is inside the polymer are repulsive and rapidly increase as the probe approaches the mica surface (where the beads comprising the polymer are constrained). These repulsive forces are attributed to the work pushing the polymer away by the probe during its penetration (see Movie S1 for the pushing motion). The most probable speed (vmp) of the beads is calculated by (2kBT/m)1/2, where m is the mass of beads.[23] Using an effective mass accounting for viscosity (m = 0.2 mg),[13] the most probable speed is 0.203 μm/s. Since the motion or configurational relaxation of the polymer is thus slower than the penetrating speed, the probe must displace the polymer out of the way during its penetration. In Figure b, we can see several vertical lines (for example, x = −32.5 and 50 nm). The movies in which these lines were observed show that the penetrating probe drags the fiber down for a distance where the lines exist (Movie S3 and Movie S4). The probe was also found to push the same parts of fiber down, while the lines are seen (Figure S1). Accordingly, the vertical lines reflect such “dragging”. This is also because of the slow relaxation of the polymer: there is insufficient time for the polymer to avoid the penetrating probe.

Figure 2

Cross sections of the polymer and simulated 3D-AFM images at vscan = 1 μm/s. (a) The structure of the polymer and four horizontal planes showing the heights of cross sections. (b) An xz-slice (at the center of the polymer) of a simulated 3D-AFM image computed using the Jarzynski equality. Scanned area was 500 nm × 450 nm with 201 × 91 pixels. (c) Color key for the force. (d) Cross sections of the polymer. (e) Four xy-slices of the simulated 3D-AFM image computed using the Jarzynski equality. Scanned area was 500 nm × 500 nm with 201 × 201 pixels. (f) Four xy-slices of the simulated 3D-AFM image computed using thermodynamic integration. Scanned area was 500 nm × 500 nm with 201 × 201 pixels.

The sliced images of the polymer at various heights are shown in Figure d. The heights where the polymer was cut are indicated by planes in Figure a. The relevant xy-slices of the 3D-AFM image are shown in Figure e. Figure S2 indicates selected parts where fiber structures match 3D-AFM images well, and Movie S5 shows the superimposed movie of Figure d,e. Again, the forces on the probe outside the polymer are attractive and repulsive inside. In the xy-sliced images, fiber structures are seen in some places. Because of the absence of dragging in the xy-direction, there is no vertical (y-direction) or horizontal (x-direction) lines in the xy-slices. Thus, the fiber structure is more clearly seen in xy-slices than in xz-slices. These fiber structures in force mapping are seen much more clearly in sequential images (Movie S6). In a comparison of these fiber structures in the force mapping (Figure e) with the polymer model (Figure d), it was found that polymer structures are indeed, but not completely, resolved in the 3D-AFM image (Figure S2 and Movie S5). The structure–force correlation is quantified later.

Figure f shows slices of the 3D-AFM image computed using thermodynamic integration. In all images, only weak attractive forces (approximately −15 pN) are seen. In this computation, the probe was stopped at a certain height and the potential energies were sampled and averaged (Movie S2). Most forces acting on the probe, which practically act on the end of the probe, were negative because the polymer and the end of the probe were located in energetically favorable positions during the equilibrium sampling process. This is the reason why attractive forces were observed.

The images computed using thermodynamic integration are completely different from those using the Jarzynski equality, since the work in a nonequilibrium process (pushing away fibers and/or dragging) was not accounted for in thermodynamic integration. Thus, no clear fiber structures are seen. Accordingly, it is significant to simulate 3D-AFM images using the Jarzynski equality in such cases where sample motion is slower than penetration speed, resulting in nonequilibrium work being performed.[15] In other words, forces measured by the AFM experiments in the biological samples whose molecular motions are presumably slower than the probe penetration are not accurately represented by the mean force.

It is known that for certain systems 3D-AFM images (or force–distance curves) change depending on the vertical scanning velocity,[19,24−26] and that the free energy profile computed by the Jarzynski equality deviates from the potential of mean force evaluated using umbrella sampling when probe velocity is fast.[27−29] As such, 3D-AFM images were additionally simulated at vscan = 0.1 (∼0.5vmp), 0.2 (∼vmp), 0.5 (∼2.5vmp), 2 (∼10vmp), 3 (∼15vmp), 4 (∼20vmp), 7.5 (∼36vmp), and 10 μm/s (∼50vmp) (see Movie S7 for a comparison of probe penetration and motions of the fiber depending on vscan). The xy- and xz-slices of 3D-AFM images at six velocities are shown in Figure a (see Figure S3 and Movie S8 for at all velocities but with different contrast). It was found that the simulated 3D-AFM images reproduced experimental observations, for example, titin unfolding.[19,24−26] The forces at vscan = 0.1 μm/s (∼0.5vmp) are so weak that they would be difficult to detect in real experiments. A detectable force is in the range 1–100 pN, depending on the measurement conditions, and is <10 pN for the ideal condition in the dynamic mode.[30,31] At this velocity, the fiber can avoid the penetrating probe, because the scanning velocity is slower than vmp and the forces are close to those obtained by thermodynamic integration (Figure b). Even at vscan = 0.2 μm/s (∼vmp), the fiber can still avoid the penetrating probe, resulting in weak forces.

Figure 3

Dependency of 3D-AFM images on the scanning velocity (vscan). (a) xy-slices at z = 250 nm and xz-slices at the center of the 3D-AFM images. Dashed magenta lines in the bottom indicate the height of xy-slice images. (b) The force–distance curves at x = −50 and y = −40 nm. Red curves are for those obtained using the Jarzynski equality, and black curves are for the thermodynamic integration during the probe approach. Blue curves show the force–distance curve while retracting. (c) Snapshots when large repulsive forces are detected.

As expected from the Maxwell–Boltzmann distribution, the speeds of almost all beads composing the fiber are less than 2.5vmp. Accordingly, when vscan ≥ 0.5 μm/s (∼2.5vmp), the motion of the fiber is slower than the scanning probe (see Movie S7), and a sufficiently detectable force acts when the probe penetrates the polymer. At vscan = 0.5 μm/s (∼2.5vmp), we can see some fiber structures in the simulated 3D-AFM image. At vscan = 1 μm/s (∼5vmp), the image becomes clearer (Figure a and Figure S3). The fiber structures are also observed at vscan = 2 μm/s (∼10vmp) and 3 μm/s (∼15vmp), while the forces increase (Figure a and Figure S3).

Figure b shows the force–distance curves. For approaching curves (red lines), we can see that the forces increase when vscan is speeded up. On the other hand, the forces computed by using thermodynamic integration (black lines) are always close to the experimental detection limit and those evaluated by the Jarzynski equality at vscan = 0.1 μm/s, since, as explained, the motions of beads comprising the fiber are faster than vscan. The force–distance curves during retraction were computed by using the Jarzynski equality and are shown by the blue lines. There is no pushing away motion of fibers and dragging in the retracting process, so the force–distance curves have, as expected, no large peak, and there is almost no dependency on vscan.

When the scanning velocity is in the range 2.5vmp ≤ vscan ≤ 15vmp, mainly two peaks are seen at ∼160 and ∼360 nm (Figure b, see Figure S4 for at all velocities but with different scale). These two peaks (i and ii) originate from dragging and pushing away, respectively (see Figure S4). Thus, they are shared among vscan = 0.5 (∼2.5vmp), 1 (∼5vmp), and 2 (∼10vmp) μm/s (also at vscan = 3 (∼15vmp) μm/s, Figure S4). Figure c shows snapshots when the probe feels a strong repulsion. Irrespective of vmp, the probe pushes away the parts of fiber colored in blue and green at the peaks i and ii, respectively. The peak i is broadened because of the dragging of a part of the fiber colored in blue (Figure S4).

Even at vscan ≥ 4 μm/s (∼20vmp), the faster the vscan is, the stronger the forces become (Figure a, Figures S3 and S4, and Movie S8). The probe, as mentioned, sometimes cuts the polymer, and the 3D-AFM images were computed before such breaks. When changing the contrast (see Figure S3 and Movie S8), the fiber structures are still seen at vscan = 4 μm/s (∼20vmp); however, there are some spots of high forces in addition to the fiber. There are cases where peaks in force reflect different fibers, and other cases originate from the dragging (Figure S4). It was found that the dragging increases when vscan is fast (Figure S5). Another reason for high forces is that the probe pushes downward not only a single fiber but also another one behind it (Figure S6). Therefore, the peaks of forces are not well-separated for each fiber. Accordingly, in the force–distance curve at vscan = 4 μm/s (Figure b, see Figure S4 for those at vscan = 7.5 and 10 μm/s), there are several peaks in addition to the two main peaks seen in the range 0.5 μm/s ≤ vscan ≤ 3 μm/s. When vscan is 7.5 (∼36vmp) and 10 (∼50vmp), the 3D-AFM images become similar to static noise and the fiber structure is difficult to see (Figure S3 and Movie S8).

The structure–force correlation (C) at each height was evaluated bywhere A, F, and Fmax are the scanned area, forces at each position in 3D space (so the 3D-AFM image), and maximum force in the entire image, respectively. σ is 1 if the position is within 15 nm from the centers of beads and −1 if not, in which the position of the beads was shifted upward by 30 nm since the force is detected approximately 30 nm above the polymer. At vscan ≥ 4 μm/s, forces before the probe cut the polymer were used for the analysis, and the area A was changed according to the number of analyzed force–distance curves.

Figure indicates the dependence of the structure–force correlation on vscan. Up to 3 μm/s, the correlation increases when vscan becomes larger (Figure a). Further increase of vscan, however, decreases the correlation because of the static noise (Figure b). Accordingly, the 3D-AFM image best matches the polymer coordinates at vscan = 3 μm/s (∼15vmp) (Figure c). Thus, there exists an optimum velocity range for vertical scan, and the force curve would reflect a rather detailed fiber structure when the scanning velocity is adequately adjusted.

Figure 4

Structure–force correlations at all vertical scanning velocities examined. (a) For vscan ≤ 3 μm/s. (b) For vscan ≥ 3 μm/s. (c) Averaged correlation against vscan.

Lastly, a comparison with the experiment in which the 3D organization of cytoskeletal fibers was visualized by 3D-AFM measurements[9] is shown. This is the best example for comparison because both the actual structure and the 3D-AFM image are known. In the 3D-AFM image of cytoskeleton fibers (Figure 3C,D in ref (9)), we can see high force regions running up and down in 3D space, reflecting the structure of the cytoskeleton fibers. Note that these images were processed to reduce the forces that originated by dragging. We mimicked the straight fiber structure by increasing the stiffness (Figure a) and computed its 3D-AFM image (see Supporting Information for details of the method). In the successive xy-slices of the 3D-AFM image from higher to lower positions (Figure b and Movie S9), regions of high force appear in sequence according to their heights as in the experiments. In the sliced image of lower position, the afterimages of fibers in the higher position appear as weaker forces because of dragging. In the xz slice of the 3D-AFM image (Figure c), elongated triangles are visible. This shape reflects the position of fibers and the dragging afterward while the nanoprobe is moving down. The triangular shape is caused by the finite diameter of the probes, which laterally touches the fiber even when it is not perfectly on the center of the fiber. The dragging strength decreases as the probe moves away from the center (Figure S8). These characteristic triangles are also seen in the experimental images before the force reduction process (Figure d). The forces computed using the thermodynamic integration are very weak (<5 pN) and are usually negative as in the case of the globular structure (Figure S9). Compared to the 3D-AFM image using the thermodynamic integration, the 3D-AFM image using the Jarzynski equality is closer to the experimental image. A remarkable difference between Figure c,d is a reversed contrast. In the simulation, the force is larger when the probe contacts the fiber and less during dragging after laterally moving the fiber. In the experiment, on the other hand, the signal is weak above the fiber and gets stronger as the probe pushes the fiber downward. A plausible explanation is that cytoskeleton fibers in cells are attached to the cell membrane, and thus, the force would become larger while the probe pushes the fiber downward due to the interaction between fiber and membrane. However, such an interaction is not implemented on the simulation. Therefore, for relatively stiff straight fibers, the method developed here shows some qualitative agreement with the experiment, although the force magnitude during dragging is different.

Figure 5

Structure of cytoskeleton fibers and its simulated 3D-AFM images. (a) Top and side views of the simulation system of cytoskeleton fibers. Scanned area is shown by the magenta box. (b) xy-slices of the 3D-AFM images at positions i, ii, and iii shown in panel a. Color key for the force is the same as in Figures and 3. (c) xz-slices of the 3D-AFM images at positions iv and v shown in panel a. (d) The experimental images of xz-slices of the 3D-AFM image of cytoskeleton fibers (left), and an xz-slice where the positions of xy-slices are shown by dashed cyan lines (right).

In conclusion, we developed a method to compute 3D-AFM images of biopolymers using the Jarzynski equality. Our simulation demonstrated that 3D-AFM technology is even capable of resolving fluctuating biopolymers: some fiber structures were clearly observed in the simulated 3D-AFM images. This supports previous 3D-AFM measurements observing molecular chains.[4,7,8] The computed forces increased when the scanning velocity becomes fast, which qualitatively reproduced force-scanning velocity relations in some experiments.[19,24−26] Moreover, the optimum scanning velocity was found in the range of several to 10 times faster than the most probable speed of the biopolymer. It was found that the use of thermodynamic integration to compute force–distance curves and 3D-AFM images is unsuitable when the motions of the samples are slower than the penetration speed of the AFM probe, because the system is far from thermodynamic equilibrium. It is expected that the motions of biomolecules are relatively slow; thus, the forces measured in the AFM measurements of biomolecules would be forces in a nonequilibrium process rather than the so-called mean force. The method developed here is applicable to various fibers in cells such as DNA and so on by changing parameters such as stiffness, providing an important theoretical base for such experimental measurements. As an example, it was applied for cytoskeleton fibers to be compared with recent experiments.[9] Another important issue for the near future is to establish a method to convert 3D-AFM images to actual fiber structures in order to utilize 3D-AFM imaging as a tool to reveal structures of biopolymers. For this, AI using machine learning or some image analysis technologies are expected to help.