ABSTRACT

A. Simulation overview

Our simulation experiment considers the dynamics of a bead attached to a surface by using classical physics–based analysis. All the relevant parameters in this model can be altered by the user to explore alternative scenarios that aid in student learning. Further, parameters that are variable in the actual SMFS experimental setups are designed to be dynamic and can be modified by the user in real time during the simulation, mimicking an actual experiment being conducted in real time as well. The static and dynamic parameters associated with a typical single-molecule TPM system are summarized in Table 1.

Table 1.Parameters considered in the TPM simulation. Static refers to fields that are assigned before the TPM simulation starts. Dynamic refers to fields that can be modified during the simulation.

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Based on these parameters, a complete description of the tethered bead position, applied force, intrinsic force due to particle collisions, energy, and tether extension from equilibrium are provided in the form of continually updated graphic plots in a MATLAB-based GUI. These plots are updated at a rate specified by the user in a static field prior to start of the simulation. The interface in which each of these parameters are provided by the user and the key features of the simulation package are summarized in Figure 1.

B. Simulation logic

A single MATLAB function that accepts user-generated parameter space, as well as memory terms, was used. This function is called within a loop in the MATLAB App Designer, and the outputs are plotted at the user specified rate. Callback functions are used in the interface to synchronize the point at which the user makes a change and when that change is reflected in the base code output. The use of memory terms in the app allowed for the computations to be done with continuity, as the user inputs are monitored and updated continuously within the base algorithm. Several simulations of the length specified by the user are run consecutively with initial conditions consistent with the end state of the prior simulation. This results in a continuous generation of data until the user-specified total number of data points are reached. A flow chart of the simulation logic is found in the Supplemental Figure S1. The simulation produces an output file in a comma-separated value (csv) format that contains time (s), planar position (m), net DNA force (N), applied force in z (N), Cartesian DNA force components (N), and θ/φ angular positions (-).

C. Computational framework of the simulation

The notations outlined in Tables 1 and 2 will be used to reference each variable in this work.

Table 2.Summary of parameter notations used for the MATLAB code and variable descriptions with reference to the defining equation (if applicable). Tether extension (r) and DNA force (net) are the vector quantities of the position and DNA force in x, y and z, respectively.

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The modified Marko–Siggia wormlike chain model was considered for our model in Eq. 1 (25). Numeric root finding was used to solve for the approximate magnitude of the force for each direction. (1)$$F_i=\left(\frac{k_{B}T}{4L_p}\left[\frac{1}{{(1-\frac{R_i}{L_o}+\hspace{0.17em}\frac{F_i}{K_o})}^2}-1+\hspace{0.17em}\frac{4R_i}{L_o}–\frac{4F_i}{K_o}\right]\right),\hspace{0.17em}i=x,y,z$$

In this model, the $\frac{F}{K_o}$ terms are a correction introduced to the classic wormlike chain model to account for the elasticity of the tether. This modification improves the experimental agreement of the wormlike chain model that provides only an order of magnitude estimate of the persistence and contour lengths (26). The $K_o$ term is a material parameter described by the Young modulus from classical mechanics. In this simulation, the Young modulus was related to the persistence length of a solid rod with a circular cross section for mathematic simplicity (25). The DNA diameter of d = 1.6 nm was chosen, and the Young modulus depends on the user-defined temperature and persistence length in Eq. 2. Some typical values of this parameter range from 800 to 1,700 pN (27, 28). (2)$$K_o=\left(\frac{16k_{B}T}{d^2}\right)L_p$$

A spherical coordinate system was used to describe the particle’s 3-dimensional motion. The obtained magnitude was decomposed into x, y, and z components via projection onto a Cartesian system by using the following elementary trigonometric relations in Eqs. 3–5. (3)$$F_x=F^{\text{*}}\left|\text{sin}{\theta }^{\text{*}}\text{cos}\phi \right|$$ (4)$$F_y=F^{\text{*}}\left|\text{sin}{\theta }^{\text{*}}\text{sin}\phi \right|$$ (5)$$F_z=F^{\text{*}}\left|\text{cos}\theta \right|$$

The signs of these quantities were determined directly through the consideration of the extension of the tether. If the tether was extended in a negative direction, the force would have to be positive to restore the system to its equilibrium position and vice versa. This behavior is consistent with classical spring behavior described by Hooke law and serves as a reasonable description for the behavior of the tether–bead system at any point in its motion due to the elasticity of the tether. All these computations were completed in a MATLAB function named MarkoSiggiaVectorized.m, and these force computations were continuously updated in a loop from the base code. The Supplemental Material documentation of the simulation package provides greater detail on the functional dependencies.

The computation of the θ and φ positions also come from basic trigonometric relations in Eqs. 6 and 7. The spatial orientation of the system is initially defined to be along the Cartesian z direction alone, and the descriptions of the angles are updated as the motion evolves over time. (6)$$\theta ={\tan }^{-1}\left(\frac{\sqrt{x^2+y^2}}{z}\right)$$ (7)$$\phi ={\tan }^{-1}\left(\frac{y}{x}\right)$$

Next, the function TetherForce2.m base code modifies the implicit force term in the z direction based on the magnitude of external force applied to the system in the user interface. The first option for the user is to choose a modifiable but constant force at any point in the simulation. When the user makes a modification to the applied force using the force slider built into the app, a constant force is continually applied to the system for the duration that the user leaves the slider in the given position. This force is immediately applied with the chosen magnitude. The second option for the user is to apply a force ramp with a slope predetermined by the user. The desired force by the end of the simulation is computed to increase in linear increments consistent with the total run time of the simulation. The third option for the user is to apply a decaying force ramp that is computationally equivalent to the previous case, except that a linear decay is considered instead. The projection of this magnitude onto the z direction is added to the force term from the MarkoSiggiaVectorized.m function to determine the net force such that $F_{\text{net}}=F_z+F$_applied, considering the force decomposition based on the second law of Newton.

The net effect experienced by the bead is intended to be consistent with Stokes law. The bead is assumed to be perfectly spherical, surfaces are all assumed to have no imperfections, components are all assumed to be entirely homogenous, and the flow is constrained to be laminar. This means that the system has a low Reynolds number that is consistent with smooth and constant fluid motion (i.e., laminar flow). When the Reynolds number is low, viscous force is necessarily dominant, meaning perturbations introduced by the bead on the system are not the variable that dominates the overall motion. The liquid viscosity is a crucial variable in determining the scale of such effects, and a specific analysis of relevance of the viscosity is presented in Supplemental Figure S3. A numeric validation of these assumptions is provided in the validation section of this study. Because all these conditions are approximately valid in the considered model, Stokes law serves as a reasonable approximation to the net effect that is observed. This also means that enough information is available so that the deviations in position within a given timestep can be extrapolated from the simulation, as is outlined in Eqs. 8–10. A correction factor is introduced to account for the edge effects because the tether–bead system is near the surface throughout the simulation. These corrections are derived based on the boundary condition that tangential flow needs to be zero at the bead surface (29, 30). The x displacement (Δx) is parallel to the surface and is described in Eq. 8. (8)$$\Delta x\approx \frac{F_x\Delta t}{6\pi \mu R}\text{*}\frac{1}{1-\frac{9}{16}\left(\frac{r}{z}\right)+\frac{1}{8}{\left(\frac{r}{z}\right)}^3-\frac{45}{256}{\left(\frac{r}{z}\right)}^4-\frac{1}{16}{\left(\frac{r}{z}\right)}^5}$$

Similarly, the expressions for the y and z directions can also be obtained. The y displacement (Δy) is parallel to the surface, so the correction due to surface effects remains the same. (9)$$\Delta y\approx \frac{F_y\Delta t}{6\pi \mu R}\text{*}\frac{1}{1-\frac{9}{16}\left(\frac{r}{z}\right)+\frac{1}{8}{\left(\frac{r}{z}\right)}^3-\frac{45}{256}{\left(\frac{r}{z}\right)}^4-\frac{1}{16}{\left(\frac{r}{z}\right)}^5}$$

The displacement in z (Δz) is perpendicular to the plane, so the correction due to surface effects is slightly different. This correction results in Eq. 10: (10)$$\Delta z\approx \frac{F_z\Delta t}{6\pi \mu R}\text{*}\frac{1}{\left(1-\frac{9}{8}\left(\frac{r}{z}\right)+\frac{1}{2}{\left(\frac{r}{z}\right)}^3-\frac{57}{100}{\left(\frac{r}{z}\right)}^4+\frac{1}{5}{\left(\frac{r}{z}\right)}^5+\frac{7}{200}{\left(\frac{r}{z}\right)}^{11}-\frac{1}{25}{\left(\frac{r}{z}\right)}^{12}\right)}$$

Here, Eqs. 8–10 are used to update the x, y, and z position at a given timestep. These Cartesian position elements are then used in Eqs. 6 and 7 to update the spatial orientation elements from the initial state due to the viscous motion.

The rearrangement of Stokes law is only an approximation because finite timesteps are used to approximate the velocity of the bead and particle in addition to the numeric approximation of the surface effects. However, an effort was made to more accurately account for time that the particle takes to move between any 2 given positions through the introduction of a dynamic timestep (30) described in Eq. 11. (11)$$\Delta t=\frac{2\mu \delta R}{\left|\nabla F\right|}$$

The implementation of this dynamic timestep allows for the accuracy of the position at any given timepoint to be the same, because the deviation is normalized using the force gradient at every data point (30). The modified Marko–Siggia model accounts for the extensibility of the tether. This has a direct influence on the dynamic timestep that depends on normalization using the force gradient. The explicit computations are shown in Eqs. 12 and 13. (12)$$\nabla F=\frac{\frac{2}{L_0{d}_x}+\frac{4}{L_0}}{\frac{4L_P}{k_{B}T}+\frac{2}{K_0{d}_x}+\frac{4}{K_0}},\frac{\frac{2}{L_0{d}_y}+\frac{4}{L_0}}{\frac{4L_P}{k_{B}T}+\frac{2}{K_0{d}_y}+\frac{4}{K_0}},\frac{\frac{2}{L_0{d}_z}+\frac{4}{L_0}}{\frac{4L_P}{k_{B}T}+\frac{2}{K_0{d}_z}+\frac{4}{K_0}}$$ (13)$$d_i={\left(1-\frac{i}{L_0}+\frac{F_i}{K_o}\right)}^3,i=x,y,z$$

After these predictable effects are accounted for, the last remaining component necessary for accurately describing the particle position is its random motion due to diffusion. The correction factor to the position predicted from the basic force analysis was implemented using a random number generator. The random number generator was set so that the mean value is zero and a standard deviation defined by Eq. 14. The environment is assumed to be approximately isotropic, as previously mentioned, which means that the expected standard deviation is independent of direction. (14)$$\sigma ={\sigma }_x={\sigma }_y={\sigma }_z=\sqrt{2\Delta \text{t}\frac{k_{B}T}{6\pi \mu R_b}}$$

The assumptions outlined previously allowed for a complete description of the position of the bead–tether system to be generated under applied force. Because the time associated with motion between any 2 given positions is also available, many useful computations can be done to test the validity of the code (for example, the verification of the conservation of energy). The general relation $F=-\nabla \text{U}$ was considered. The force acting on the particle is approximated to be constant and independent for a given timestep. This means that the potential energy can be approximated using Eq. 15. The force terms are constant in each interval; thus, the integration only occurs over the position differential. (15)$$\Delta {\text{PE}}_i\approx -(F_x\text{*}\left(x_i-x_{i-1}\right)+F_y\text{*}\left(y_i-y_{i-1}\right)+\hspace{0.17em}{F}_z*\left(z_i-z_{i-1}\right))$$

The projections of the force on the x, y, and z direction and the displacement as calculated between subsequent timesteps are considered in determining the change in potential energy. The kinetic energy was computed and rewritten using parameters relevant to the constructed system in Eq. 16. (16)$$\Delta {\text{KE}}_i\approx \frac{2\pi \rho R_b^3}{3}\text{*}\frac{{\left(x_i-x_{i-1}\right)}^2+{\left(y_i-y_{i-1}\right)}^2+{\left(z_i-z_{i-1}\right)}^2}{{\left(\Delta t\right)}^2}$$

The ρ term in the kinetic energy computation is the density of the bead, the displacement in each direction is determined at subsequent timesteps iterated by variable i, and the length of the timestep is denoted $\Delta t$.

All the arguments made previously are consistent with a probabilistic consideration of the behavior of the tether–particle system. This means that by nature, assumptions of thermal equilibrium are made, as can be seen from the applications of the equipartition theorem. These assumptions are not appropriate when a force is instantly applied to the system. The force ramp feature used allows for a steady buildup of the force that does not perturb the system to a great extent at any given instant. However, for general applications of large magnitudes of force, this model can break apart. As such, a separate model was implemented as described in the following, which uses physical constraints to ensure that the system remains stable, as expected in reality.

First, in the cases in which an external force is applied to the system, it is approximated that $F_{\text{net}}\approx F_{\text{applied}}$. When a force is applied, the tether will extend, and the tension will increase resulting in limited fluctuations. In accordance with the modified Marko–Siggia model (25), these fluctuations will occur with an equilibrium value that is associated with the applied force. Extracting this equilibrium value allows for a description of the bead position to be made independent of the timescale associated with the instability. Generating a distribution of permissible values about this equilibrium position allows for a complete description of the bead position. It is constrained by the tether length, and there is a very small probability the bead will reach a value significantly different from the equilibrium value. These constraints are well described by a normal distribution with 0 mean fluctuations about the equilibrium position. The standard deviation was determined, as shown in Eq. 17, where $\alpha$ is an arbitrary parameter meant to describe the resistivity of the environment and z_eq is the extracted equilibrium position. This parameter is not strictly defined and can be modified to best fit the data collected. In accordance with the assumption of isotropy, $\alpha =3$ was assigned as the base setting. (17)$${\sigma }_z=\frac{L_o-z_{\text{eq}}}{\alpha }$$

Because the z spatial harmonic behavior is described, the planar region of interest can easily be extrapolated. The system is defined so that magnitude of the position vector corresponds to the tether extension. Because a reasonable approximation to the z position is obtained, the acceptable x and y positions must be approximately consistent with the constraint in Eq. 18 because a force regime in which unwinding of the double-stranded DNA occurs (∼65 pN) is not considered here. (18)$$x+y=\hspace{0.17em}{L}_o-z_{\text{eq}}$$

The distribution of the x and y positions are not expected to have significant bias because an isotropic environment is considered. As such, the weight of the permissible positions will be approximated to be equivalent.

Using these 2 conditions, a constraint for the x and y position can be obtained. Because the viscous effects also have a contribution to the planar variations, a distribution was generated under the assumption of thermodynamic equilibrium in Eq. 14. The instantaneous application of force in the z direction is accounted for without the consideration of thermodynamic equilibrium. Essentially, this means that the system is forced into a harmonic state that can be described by conditions using thermodynamic equilibrium. The implementation of the previously described method is a boundary condition that restabilizes the environment. This means that the assumption of thermodynamic equilibrium is valid after the system is constrained with this method. This equilibrium is artificial in the sense that constraining the system requires higher force fluctuation magnitudes than is experimentally observed. A spatial resolution of 10% was used to limit the simulation to values similar to experimental fluctuations; Eqs. 19 and 20 describe the x and y positions of the bead within this framework. The computations done in the simulation begin by considering the origin of the system in the frame of the bead. All the terms were rescaled so that the result is consistent with these relations where x_visc and y_visc are Brownian terms that account for the viscous motion and possibilities for extension of the tether. (19)$$y=\frac{(L_o-z_{\text{eq}})}{2}+y_{\text{visc}}$$ (20)$$x=\frac{(L_o-z_{\text{eq}})}{2}+x_{\text{visc}}$$

The assumptions made when generating this simulation are consistent with the assumptions made in a typical Markov process (31). The distributions from which the Brownian fluctuations are determined are normal with a mean of 0 and standard deviation defined in Eq. 14. All Brownian fluctuations are extrapolated from probability distributions governed by the same rules, are time independent, and intrinsically constrain how far a particle could be displaced due to a collision with the molecules in the viscous environment. This means that at any given position, the span of reasonable values classically obtainable by the particles is predefined for a given timestep. Finally, each state attained by the particle is assumed to be independent of every subsequent state obtained by the particle. This is consistent with the constraints associated with classical Brownian motion (32).

Within the simulation, there are 2 models implemented. The first, which the simulation is initialized to use as the toggle switch, is the trend model. This model is intended only for educational purposes. It is a mixture of both model types mentioned previously, wherein the point at which the first model breaks down is point in which the new model is implemented. In other words, in absence of force, the probabilistic model, described by Eqs 1–16, and the constraint model, with Eqs 17–20, are both implemented. The first model is purely probabilistic in nature and gives predictions closer to the equilibrium state on average, whereas the second model is much stricter, and the values accepted tend to be more confined. In transitioning between these models, physically unrealistic results may be occasionally observed. However, there are many benefits to this model, as described in the following discussions. For stricter data collection mode, the toggle switch must be switched to data, where the numeric inconsistencies from the force application will not exist.

A. Implementation and theoretic validation

All data are generated using 2 primary tiers of code. First, TetherForce2, runs the simulation based on the equations previously described for using loops. Next, a loop in the MATLAB App Designer was used to continually update the arrays containing the parameter space generated. The results generated in the base code based on the updated parameter space are immediately assigned to relevant GUI axes to plot the results, as the simulation continues running in real time. This second tier of code is the most inefficient component of this simulation because it must check for user input at every data point, update the parameters the base function calls at every data point, and plot the data at a rate specified by the user. A more comprehensive discussion of the simulation efficiency is provided in the Supplemental Appendix. In case the code is being used solely for data generation (and not GUI-based results for visualization in real time), the user can set the plot rate equal to the total number of data points, and this will result in significantly improved run time. Although slightly more computationally intensive, it was found that continual application of a force did not significantly affect the run time of the simulation.

Aside from the actual implementation, the simulation gives a reasonable approximation to the physical behaviors associated with a typical tethered particle–bead system. The tether particle–bead system will display a wide range of fluctuations in every direction provided that an external force is not applied to the system. The parameter space used to generate the results plots are outlined in Table 3.

Table 3.Sample parameter space used to generate Figure 2 result plots. The parameters are approximately consistent with the simulation parameters considered in the implementation by Beausang et al. (41).

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Due to the lack of significant tension on the tether in the absence of an applied force, the planar variation will be more prominent because the tether–bead system will not have any rigidity, as shown in the upper panel of Figure 2.

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In the presence of an applied force, it is expected that the particle will asymptotically approach maximum extension over a given time and the x–y motion will be confined to smaller scales due to the tension exerted by the tether. The lower panel of Figure 2 was generated using the trend mode of the simulation. The details of this mode will be discussed in more detail in the following.

In the absence of an applied force, the planar position of the particle is unrestricted and varies with a span of approximately ±800 nm. In the presence of an applied force, the span in which the planar position varies is limited to approximately ±200 nm. This behavior becomes more evident as the magnitude of the force increases over time because the tether will experience increasing tension. In presence of an applied force, the system asymptotically fluctuates near maximum extension. All these behaviors are consistent with theoretic expectations.

The final features implemented into this simulation are analysis plots. A useful analysis that validates the results of the simulation is the consideration of the applied and intrinsic DNA forces as a function of the net extension (r), presented in Table 2, which represents the magnitude of the 3-dimensional radius vector of the position for each Cartesian component (x, y, z). In this model, the maximum extension should not greatly exceed the combined length of the bead and tether at any timepoint because the force application is limited to a regime in which double-stranded DNA helix unwinding is not relevant, as previously discussed. Figure 3 confirms this physical restriction both in the presence and absence of an applied force. Further validation using energy analysis is presented in Supplemental Figure S3.

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B. Model validation via AFS experiments

For the comparison of the simulation results with SMFS experimental data, a TPM experiment was carried out in our laboratory by using an Acoustic Force Spectroscopy instrument (AFS) (LUMICKS, Amsterdam, The Netherlands). A more detailed description of the experimental setup is found in the Supplemental Appendix. Briefly, the surface of the AFS chip is incubated with anti-digoxigenin fab fragments for 20 min, followed by surface passivation with bovine serum albumin (BSA) protein, casein protein, and Pluronic F-127 nonionic surfactant in 10 mM phosphate-buffered saline solution for 30 min. Next, DNA functionalized on opposite ends with biotin and digoxigenin is mixed with streptavidin-functionalized polystyrene beads for 30 min, washed twice in PBS containing BSA, casein, and Pluronic F-127, and incubated in the AFS imaging chip for 15 to 30 min. Finally, nonbound beads are flushed out, and the remaining beads are tracked in 3 dimensions. Analysis of bead traces was performed with the software provided by LUMICKS (AFS-Analysis-G2 version, Amsterdam, The Netherlands), with slight modifications (34), and a free academic version can be found in the original publication of the AFS (22).

As can be observed in Figure 4, the simulated root-mean-square (RMS) position values of the bead–tether systems are of the same order of magnitude and follow the same trend as the AFS experimental values, based on the simulation parameters presented in Table 4. The RMS value (22) was determined by using Eq. 21, where $\overline{x}$ and $\overline{y}$ are the average x and y positions, respectively, of the position coordinates presented in Table 2. (21)$$\text{RMS}=\sqrt{{\left(x-\overline{x}\right)}^2+{\left(y-\overline{y}\right)}^2}$$

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Table 4.Sample parameter space used to generate Figure 4 result plots.

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As the magnitude of applied force increases, the equilibrium RMS position the system takes decreases. The discrepancy in experimental RMS values is due to experimental limitations such as uncertainty in the exact bead size, resolution of the AFS instrument or technique, and model limitations. Namely, an order of magnitude approximation is being conducted on the basis of previously described assumptions, so while the trends are captured, the model itself does not have a resolution allowing better certainty to be achieved for all types of analysis. In these simulations, the bead diameter was set at 3,110 nm. The bead diameter does not influence the harmonic behavior and affects only the scaling of the time from the earlier described dynamic timestep. The experiments were carried out in phosphate buffer supplemented with low concentrations of proteins and polymers (see Supplemental Material); however, it is assumed that those additives did not change the viscosity (35). Thus, the simulation uses the viscosity of pure water.

C. Case study demonstrating use of the GUI for TPM modeling in a classroom setting

The following section outlines the question that could be asked by an educator, implementation of the TPM simulation tool kit to address the question, and specific steps taken within the tool kit GUI to obtain a suitable answer generated by the students. One example question posed by the educator to the students could be “How does the time averaged <RMS> value vary as a function of tether contour length in TPM?”

Steps taken by the students or instructor to address this specific question are briefly outlined in the following:

• (a) Open the simulation and show the students each of the variable meanings using the definition tab. Emphasize the importance of fixing all variables, except the tether length, because that is the variable of interest.

• (b) Assign physically reasonable values (such as the default values) to all the variables, except the tether length, consistent with the type of system that is being studied.

• (c) Choice 1: The simulations could be run prior to the lecture, and the output files could be displayed in a suitable dataset format to the instructor’s preference.

  Choice 2. The class can be divided into groups, and each group can run a simulation for a particular tether length. The results can then be combined for the final plot.

  The simulated data are saved in csv format in the same folder as the GUI, which contains labeled fields, including the time (s); x, y, and z position data (m); the total force (N); the applied force (N); the x, y, and z components of the force (N); and the θ/φ angular positions.

• (d) When the simulation is completed, the script Example_1_script included with the sample lesson plan can be used to automatically output the RMS values for a given simulation.

• (e) The general trends could be displayed through the creation of plots, as shown in Figure 5. Notes about the nature of Brownian motion and the consequent variation in results among trials could be made. To account for such variability, the trials were conducted 3 times per tether lengths, and an average was obtained to generate the figure.

• (f) The characteristic behavior in which the RMS value tapers off, as tether length increases (27), should be highlighted by the instructor. Limitations regarding the models in general should be discussed with students.

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The raw data used to generate Figure 5 in csv format can be found on GitHub (https://github.com/ChundawatLab/TPM-GUI), along with an example to create and analyze force–extension curves, summarized in a lesson plan. The GitHub material also includes the GUI source code and user guide and manual. Further similar experimental comparisons are included in Supplemental Figures S5–S7. Other points of discussion could be how the behavior of these curves depends on the bead radius, which also tends to be varied in real SMFS experiments. In general, hands-on exploration of the simulation tool kit permits many modes of analysis relevant to actual experiments and provides results that are consistent with the actual trends from real-world experiments so that student learners can gain deeper insight into the concept of TPM.