ABSTRACT

I. INTRODUCTION

Students without a strong theoretical physics or chemistry background encounter the term “phase transition” in their first biophysics or physical chemistry course. A first-order phase transition is associated with the heat capacity, typically defined at a given temperature as a first partial derivative of the enthalpy (constant pressure heat capacity) or internal energy (constant volume heat capacity). Students can measure the heat capacity of various materials in the laboratory by using a calorimeter as part of a laboratory exercise during a full-fledged bachelor’s-level course in physical chemistry or fundamental biophysics. In biology, such a material could be a folded protein, a single-component or 2-component lipid membrane, or a DNA double helix. Performed carefully, such an experiment can measure the peakof the heat capacity of the material as it approaches a temperature-induced phase transition. Although this approach can reveal macroscopic properties of the phase transition, the underlying microscopic behavior of the material that led to the phase transition is often left completely unexplored. Students gain no insights into the soft forces that drive the assembly and evolution of biological macromolecules. Molecular dynamics (MD) simulations can be used to explore such microscopic behavior because the simulations enable calculation of the macroscopic properties (such as the heat capacity) while simultaneously providing atomic-resolution information on the molecular interactions in the biological material.

In this article, we describe a computer laboratory for a 2nd-year Physical Biochemistry course in which we use MD simulations to explore the folding-unfolding transition of a poly-alanine Ala₃₀ molecule in vacuum, with the objective of establishing the link between the heat capacity of the phase transition (macroscopic thermodynamic property) and the molecular details accompanying the phase transition.

A key challenge in running such a laboratory is that bachelor’s students with a background in biochemistry or molecular biology often have no prior knowledge of MD simulations or training in statistical mechanics. To circumvent these challenges, we use a graphical user interface and web platform: Scandinavian Online Kit for Nanoscale Modeling (VIKING) (1), which sets up an MD simulation and analyzes the data automatically. The students encounter the actual simulation as a black box but are made familiar with the basic principles of MD simulations both before and during the laboratory exercise. In this way, the students can use MD simulations as a tool without having to undergo a rigorous initiation into statistical mechanics or numerical integration.

The rest of the article is organized as follows. In the section “Background,” we describe the basic principles of MD simulations and calculation of the heat capacity. Under “Materials and Methods,” we describe the MD method that we use to calculate the heat capacity in a research setting. Under the section “Implementation as a Computer Laboratory,” we describe exactly how the method is implemented in a computer laboratory, which lasts ~3 h. We provide as many details as possible to enable easy incorporation into a 1st or 2nd year bachelor’s-level course in biophysics or physical or biophysical chemistry.

II. BACKGROUND

The proposed simulations use the MD package NAMD (2) to model a transition from an α-helix to a random coil phase of a single poly-alanine Ala₃₀ molecule. The laboratory design and simulations presented here are inspired by Solov’yov et al. (3).

The overall idea is to numerically solve the equation of motion for each atom in the poly-alanine Ala₃₀ molecule, which in this case is the Langevin equation (4) given by (1) $$m_i\stackrel{\mathrm{..}}{\overrightarrow{r_i}}=-\frac{\partial U(\overrightarrow{R})}{\partial \overrightarrow{r_i}}-\gamma m_i\stackrel{.}{\overrightarrow{r_i}}+\sqrt{2m_i\gamma k_{B}T}\overrightarrow{\eta }(t)$$

Here, m_i is the mass of an atom and $\overrightarrow{r_i}$ its position. The first term on the right corresponds to systematic forces arising from the interactions between particles of the solute. U( $\overrightarrow{R}$ ) is the potential energy of the system that depends on the position $\overrightarrow{R}$ of all atoms. The negative gradient of $U(\overrightarrow{R})$ with respect to $\overrightarrow{r_i}$ is the force that acts on particle i. The 2nd and 3rd term on the right are responsible for coupling the particle to a heat bath. Here, γ is the damping coefficient, k_B the Boltzmann constant, T the temperature of the system, and $\overrightarrow{\eta }$ a Gaussian distributed noise term that satisfies $\langle \overrightarrow{\eta }(t)\rangle =0$ and $\langle \overrightarrow{\eta }(t)\stackrel{·}{\overrightarrow{\eta }}(t\prime )\rangle =\delta (t-t\prime )$ . We use the semi-empirical CHARMM27 force field (5) to calculate the system potential energy $U(\overrightarrow{R})$ . The CHARMM27 force field is well suited for MD simulations of large biomolecular systems such as proteins, peptides, lipids, and nucleotides.

The solution of the Langevin equation is carried out in small time steps. In these time steps, the various derivatives of the Langevin equation are assumed to be constant, enabling discretization of the equation. By repeatedly solving the discretized Langevin equation, the dynamical behavior of the poly-alanine Ala₃₀ molecule is simulated. Effectively this is done by using the velocity Verlet algorithm, which is described in the appendix of Swope et al. (6). The system energy E(t) can be accumulated as a function of simulation time.

The heat capacity C_V of the system can then be calculated from the time averaged values $\langle E\rangle$ and $\langle E^2\rangle$ of the system energy by considering heat capacity as a derivative of the mean energy with respect to temperature T: (2) $$C_V(T)=\frac{\partial \langle E\rangle }{\partial T}=\frac{1}{k_B{T}^2}\frac{\partial }{\partial \beta }\left(\frac{1}{Z}\frac{\partial Z}{\partial \beta }\right)=\frac{\langle E^2\rangle -{\langle E\rangle }^2}{k_B{T}^2}.$$

Here, $\beta ={(k_{B}T)}^{-1}$ , with k_B being the Boltzmann constant, and $Z={\displaystyle {\sum }_n\exp }(-E_n\beta )$ is the partition function, which can be shown to be related to the mean value of the potential energy as $\langle E\rangle =-\frac{1}{Z}\frac{\partial Z}{\partial \beta }$ and $\langle E^2\rangle =\frac{1}{Z}\frac{{\partial }^{2}Z}{\partial {\beta }^2}$ . The simulation should be repeated at different temperatures, to characterize the thermodynamic properties of a phase transition, by estimating $\langle E\rangle$ and $\langle E^2\rangle$ for each simulation to calculate the heat capacities for the corresponding temperatures.

Unlike in statistical mechanics, the pseudo phase transition is macroscopically observed as a peak in the heat capacity when the heat capacity is plotted against temperature. This is in contrast to the statistical mechanical description of a phase transition of a system containing a very large number of degrees of freedom, in which the heat capacity diverges around a critical point (7). The finite peak of the heat capacity in the simulations manifests because of the finite size and sampling of the system. This is in general a limitation in the investigation of systems around a critical point using numerical methods that can be used only to investigate finite-sized systems. The peak of C_V is also rather wide because the transition from an α-helix to the random coil phase occurs gradually as temperature increases, meaning that during the phase transition, the poly-alanine Ala₃₀ molecule retains some α-helix-like and random coil flavor. The teacher may choose to emphasize these differences depending on the educational background of the students.

III. MATERIALS AND METHODS

To calculate the heat capacity of an alanine polypeptide, a number of steps must be carried out by using different computational tools.

First, the initial 3-dimensional coordinates of the poly-alanine Ala₃₀ molecule must be defined. The equations of motion, Eq. (1), are second-order differential equations that require initial positions of all particles to be specified to permit a solution. The coordinates of a perfect α-helix poly-alanine Ala₃₀ molecule were generated by using PyMOL (Fig 1) (8).

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Second, the Langevin equations must be solved numerically by using NAMD. NAMD calculates the total energy of the system and several other properties and exports them to output files. To run NAMD, an input configuration file is generated. The configuration file points to the coordinates of the poly-alanine Ala₃₀ molecule and to the CHARMM27 force field and specifies various other simulation parameters such as the temperature, the number of integration time steps, the size of the time step, and the frequency (number of time steps) with which total system energy is written to an output file.

The MD simulation is performed in 3 stages: (i) 100 steps of energy minimization, (ii) 1 ns of an equilibration simulation with an integration time step of 2 fs, and (iii) a production simulation of 200 ns. Note that the simulations performed for this article were equilibrated in a 20-ns simulation. The total energy of the system is sampled every 2,000 time steps during the production simulation. Note further that these timescales are nonphysical and will be discussed further in the discussion. More frequent sampling of data can be time consuming, and we want to avoid sampling highly correlated data points, which would not significantly improve the estimate of the mean of the total system energy.

The simulations were conducted on central processing unit (CPU) nodes with Intel Xeon Platinum 8358 32-core CPUs. Multiple cores per simulation decreased the wall time required per simulation up to about 8 cores, after which the required wall time increased as more cores were used. Because of the relatively small system size, it was optimal to run 1 simulation per CPU core, allowing for many simulations to be run simultaneously on each CPU node. It took ~20 h to finish a simulation on one core, meaning that 32 simulations could be completed per CPU within a day.

It is advisable to benchmark the performance of the simulations on the available hardware, such that an optimal distribution of computer resources can be made during the actual class (see the next section).

Significant computer time is required for running 501 simulations: 1 for every temperature value between 500 and 1,000 K. It is advisable to let each student or group of students run simulations for a small, predefined set of temperatures such that all the students doing the exercise can collect their data in a shared spreadsheet and collectively make a high-resolution plot of the heat capacity as a function of temperature. The teacher can provide the other data points a priori in the Excel sheet.

IV. IMPLEMENTATION AS A COMPUTER LABORATORY

V. DETAILED WORKFLOW

In the following sections, we illustrate in detail the workflow that the students experience. The class, typically 25 students, is divided into groups of 2–3 students. Each group is assigned to perform simulations at one temperature value to reconstruct the temperature dependence of the heat capacity curve, ensuring that the three main regimes of the unfolding transition (folded state, unfolding transition, unfolded state) are captured. Individual groups upload their data on a shared spreadsheet. The compiled data in the document can then be used to make the final calculations and construct a report. Each group submits an individual report.

A. Generation of initial coordinates by using PyMOL

The coordinates of the poly-alanine Ala₃₀ molecule can be created by following these steps:

1. Open PyMOL.

2. On the menu bar at the top of PyMOL, select Build then Residue and finally Helix.

3. Now enter fab AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA, ss=1 in the PyMOL command input and hit enter, at which point a poly-alanine Ala₃₀ molecule should appear in the view.

4. To center the newly created structure in the view window, select Display, Zoom, and finally All from the top menu bar, as seen in Fig 2.

5. The representation of the structure can be changed by finding the ala or obj01 molecule in the object overview on the right of PyMOL and clicking the A and selecting preset and finally pretty as seen in Fig 3.

6. The coordinates of the molecule can now be exported to the PDB format by selecting File and Export Molecule from the top menu. A new window should open. In this window, set the Selection drop-down to ala or obj01 and click Save. Select a location to save the file, give it a name, and set the file type to PDB.

7. To check that the exported file represents the appropriate molecule, open it with any text editor and verify that it contains 300 atoms corresponding to 30 residues. Make sure not to edit the file, to avoid corrupting it.

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B. Test simulation with VIKING

Now that the coordinates of the poly-alanine Ala₃₀ molecule are ready, the simulation can be started. The following steps illustrate how this can be accomplished by using VIKING. Note that refreshing the web page while setting up the simulation may require the process to restart from the beginning.

1. Open a Web browser and navigate to https://viking-suite.com/, read the privacy policy, and accept the cookies. Enter the username and password supplied in the class in the sign-in form at the top of the web page.

2. Go to Projects and select the project with the temperature assigned to your group.

3. Create a new task by clicking the New task button above the project overview. Select Molecular Dynamics and then Equilibrium MD. Name the tasks (e.g., MD simulation, Ala30, 450K). Go to the next step by clicking the next button on the bottom right.

4. Click the new structure button above the project overview, click the Browse button and select the PDB file exported from PyMOL, and finally click Load. The structure can be previewed by hovering the cursor over it in the project overview and clicking the three horizontal bars on the right. Go to the next step.

5. Deselect the Water and Proteins potentials at the top by clicking on them, and then select the CHARMM27 potential by clicking on it under the custom potentials in projects at the bottom. Go to the next step.

6. Choosing a solvent: Select - None - because the poly-alanine Ala₃₀ molecule does not need to be immersed in a solvent. Go to the next step.

7. Set the simulation time to 2 ns for an in-class demonstration to ensure that the simulation is set up correctly. Before launching the test simulation, the teacher should check the simulation setup. Once this is done, go to the next step.

8. Set the temperature assigned to your group. Fast forward to the final step by pressing the double arrows to the left.

9. The screen that emerges provides an overview of all the simulation parameters; at the bottom, uncheck the Use Periodic Boundary Conditions check box (Fig 4).

10. For this simulation, only two equilibration and production run stages are required. To turn off stages 1 and 2, go to their pages by clicking them at the left panel and un-checking Do stage 1 and Do stage 2 at the top of each page.

11. To speed-up the simulation conclusion for the in-class experiment, the final equilibration stage is also disabled. Navigate to the configuration page of stage 3 and un-check Do Stage 3 at the top of the page. For the longer simulation, stage 3 should be executed. The structure generated by using PyMOL must be minimized and equilibrated for the final simulation per standard MD protocols. For this, set Minimization Steps to 100, the Simulation Length to 1 ns, Restrain Hydrogen Bonds to all. Here, 1 ns is the equilibration period (Fig 5).

12. Go to the overview of Stage 4 and check that the Simulation Length and temperature are set to the correct values.

13. Go to Electrostatic options and set Cutoff Distance, Switching Distance, and Pairlist Distance to 1000 Å. Finally, un-check the Use PME option.

14. Finally, hit Run task, and the task will be created. The test simulation of 2 ns will be launched immediately if a reservation on the cluster computer has been made in advance.

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E. Visualizing data in PyMOL

The psfgen.psf and stage4_1.dcd files downloaded from VIKING permit visualizing the dynamical development of the poly-alanine Ala₃₀ molecule in PyMOL.

1. First open PyMOL and select File and Open … . Navigate to the psfgen.psf file downloaded from VIKING, select it, and press Open.

2. Select File and Open … again and navigate to the stage4_1.dcd file and select Open, which opens a new window shown in Fig 6. Click load.

3. Center the structure and set the representation to pretty as done previously in Section V.A. The dynamic evolution of the poly-alanine Ala₃₀ molecule can now be seen by pressing the play button at the bottom of the object overview at the right (Fig 7).

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F. Uploading data to the shared spreadsheet

The heat capacity values obtained from the very short in-class simulation of 2 ns are rather inaccurate, as seen in Fig 8a. However, the students are still instructed to upload their data in the shared document to ensure that a preliminary, albeit inaccurate curve is obtained from the test simulations.

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VI. RESULTS AND DISCUSSION

Student groups simulate the peptide at 8–12 different temperatures, depending on the number of students. In this article, we have performed the simulation 501 times for every integer step from 500 K to 1,000 K. Fig 8b shows the heat capacity calculated for each temperature. Students are encouraged to superimpose their data onto a similar plot after their analysis is complete.

The temperature dependence of the heat capacity in Fig 8b shows the expected behavior, with a peak starting ~640 K as hydrogen bonds begin to break and the poly-alanine Ala₃₀ molecule begins to unfold. The heat capacity has a maximum at 740 K, and the peptide reaches the fully unfolded state at 840 K. Illustrations of the poly-alanine Ala₃₀ molecule taken from the simulation at 580 K and 990 K (Fig 9) demonstrate that the peptide is unfolded at the higher temperature. Refer to the Supplementary Material for an animation showing the correlation between the temperature dependence of the heat capacity and the conformation of the peptide.

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The heat capacity calculated from the simulations is in good agreement with earlier studies (3). Still, the rather large fluctuations at low temperatures indicate that more extended simulations may be necessary for a better sampling. From a pedagogical perspective, the shortcoming of limited sampling times should certainly be made clear to the students. However, using additional computer time to decrease the magnitude of the fluctuation in the heat capacity provides no further physical insight to the students.

The simulation of the peptide is performed without a solvent to minimize computational costs associated with running a laboratory session. The results would markedly differ and be more realistic if the simulations were performed in a solvent such as water, because particles will interact differently in an explicit solvent. For example, the hydrophobic effect and electrostatic screening due to polarization of water are not captured in the current setup. These effects are further exaggerated by the lack of calibration of the coupling to the heat bath, which results in a mismatch of the timescale of the unfolding transition and temperatures at which the transition happens. Usually folded proteins unfold in water upon heating beyond 340 K. For a discussion of the folding timescale of proteins, see, for example, Kubelka et al. (14). However, the concepts and the physics of calculating a heat capacity of a phase transition is in no way undermined by using no solvent (9).

There are various theoretical approaches for modelling the helix to coil phase transition (15, 16). Although such statistical mechanical models capture the essence of the transition, the laboratory is not targeted to students with an advanced working knowledge of statistical mechanics. In addition, the purely theoretical approach will not serve as an introduction to biological phase transitions or to the coupling between microscopic and macroscopic properties. The proposed MD simulations demonstrate the coupling as well as introduce the use of computer simulations as a research tool. The coupling is achieved because the students have the ability to peer into the microscopic behavior of the Ala₃₀ molecule at any point in time and link it directly to a macroscopic property (i.e., the heat capacity).

VII. CONCLUSION

The simulation laboratory presented here predicts the heat capacity of the folding-unfolding transition of a single poly-alanine Ala₃₀ molecule. The computer laboratory exercise gives the students molecular insights into a biophysical system and the molecular and macroscopic definitions of the heat capacity of a first-order phase transition. The simulation laboratory cannot replace a conventional laboratory where students can, for example, investigate the phase transition of a lipid bilayer by using a differential scanning calorimeter. However, we are confident that the computational calculation provides molecular and statistical insights into a simple concept from physical chemistry, which is impossible to obtain from a macroscopic experimental laboratory alone.

As extensions of the laboratory, students can investigate the temperature and width of the phase transition as a function of peptide length.

Combined with a graphical user interface of VIKING, the simulation also serves as a good introduction for students to working with MD simulations that plays an important role in modern biophysical education and research.