Modeling the hydrophobic effect of a polymer

There are many concepts in biochemistry that are not as simple as they appear to be. These are things that tend to confuse you if you mull over them. Over the years, I have found osmosis such a thing. Another such thing is hydrophobicity. (As a physicist, I love these puzzles!)

Figure 1: More "polar" solvent on the right.

In our NSF-funded Constructive Chemistry project with Bowling Green State University, Prof. Andrew Torelli and I have identified that the hydrophobic effect may be one of the concepts that would benefit the most from a constructionism approach, which requires students to think more deeply as they must construct a sequence of simulations that explain the origin of this elusive effect. Most students can tell you that hydrophobicity is "water-hating" as their textbooks simply have so written. But this layman's term itself is not accurate and might lend itself to a misconception as if there existed some kind of repulsive force between a solute molecule and the solvent molecules that makes them "hate" each other. An explanation of the hydrophobic effect involves quite a few fundamental concepts such as intermolecular potential and entropy that are cornerstones of chemistry. We would like to see if students can develop a deeper and more coherent understanding while challenged to use these concepts to create an explanatory simulation using our Molecular Workbench software.

Andrew and I spent a couple of weeks doing research and designing simulations to figure out how to make such a complex modeling challenge realistic for his biochemistry students to do. This blog post summarizes our initial findings.

Figure 2. The radii of gyration of the two polymers.

First we decided that we would like to set this challenge on the stage of protein folding. There are few problems in biochemistry that are more fundamental than protein folding. So this would be a good brain teaser that could stimulate student interest. But protein folding is such a complex problem. So we would like to start with a simple 2D polymer that is made of identical monomers. This polymer is just a chain of Lennard-Jones particles linked by elastic bonds. The repulsion core of the Lennard-Jones potential models the excluded volume of each monomer and the elastic bonds link them together as a chain. There is no force that maintains the angles of the chain. So the particles can rotate freely. This model is very rough, but it is already an order of magnitude better than the ideal chain, which assumes a polymer as a random walk and neglects any kind of interactions among monomers.

Figure 3. Identical solvents (weakly polar).

Next we need a solvent model. For simplicity, each solvent molecule is represented by a Lennard-Jones particle. Again, this is a very rough model for water as solvent as it neglects the angular dependence of hydrogen bonds among water molecules. A better 2D model for water is the Mercedes-Benz model, so called because its three-arm model for hydrogen bonding resembles the Mercedes-Benz logo. We will probably include this hydrogen bonding model in our simulation engine in the future, but for now, the angular effect may be secondary for the purpose of this modeling project.

As with themselves, the polymer and solvent molecules interact with each other through a Lennard-Jones potential. Now, the question is: Are these interactions we have in hands sufficient to model the hydrophobic effect? In other words, can the nature of hydrophobicity be explained by using this simple picture of interactions? Would Occam's razor be good in this case? I feel that this is a crucial key to our Constructive Chemistry project: If a knowledge system can be reduced to only a handful of rules students can learn, master, and apply in a short time without being too frustrated, the chance of succeeding in guiding them towards learning through construction-based inquiry and discovery would be much higher. Think about all those successful products out there: LEGO, Minecraft, Algodoo, and so on. Many of them share a striking similarity: They are all based on a set of simple building blocks and rules that even young children can quickly learn and use to construct meaningful objects. Yet, from the simplicity rises extremely complex systems and phenomena. We want to learn from their tremendous successes and invent the overdue equivalents for chemistry and biology. The Constructive Chemistry project should pave the road for that vision.

Figure 4. Identical solvents (strongly polar).

Back to modeling the hydrophobic effect: Does our simple-minded model work? To answer this question, we must be able to investigate the effect of each factor. To do so, we set up two compartments separated by a barrier in the middle. Then we put a 24-bead polymer chain into one of them and then copy it to another. In order for them not to move to the edges or corners of the simulation box (if they stay near the edges then they are not fully solvated), we pin their centers down using an elastic constraint. Next we will put different types of solvent particles into the two compartments. We also use some scripts to keep the temperatures on both sides identical all the time and export the radii of gyration of the two polymers to a graph. The radius of gyration of a polymer approximately describes its dimension.

By keeping everything else but one factor identical in the two compartments, we can investigate exactly what is responsible for the hydrophobic effect for the polymers (or its relative importance). Our hypothesis at this point is that the hydrophobic effect would be more pronounced if the solvent-solvent interaction is stronger. To test this, we set the Lennard-Jones attraction between solvent B (right) particles to be three times stronger than that between solvent A particles, while keeping everything else such as mass and size exactly the same. Figure 1 shows a series of snapshots taken from a nanosecond-long simulation (this model has 550 particles in total, but on my Lenovo X230 tablet it runs speedily). The results show that the polymer on the right folds into a hairpin-like conformation with its two freely-moving terminals pointing outwards from the solvent, suggesting that it attempts to leave the solvent (but cannot because it is pinned down). And this conformation and location last for a long time (in fact most of the time during the simulated nanosecond). In comparison, the polymer on the left has no stable conformation or location -- it is randomly stretched in the solvent most of the time and does not prefer any specific location. I think this is the evidence for the hydrophobic effect in two senses: 1) The polymer attempts to separate from the solvent; and 2) the polymer curls up to make room for more contacts among the solvent particles (this is related to the so-called hydrophobic collapse in the study of protein folding). The second can be further visualized by comparing the radii of gyration (Figure 2), which consistently differ by 2-3 angstroms.

Note that we did not introduce any special interaction between the polymers and the solvent particles of either type. The interaction between the polymer with a solvent particle is exactly the same in both compartments. The only difference is the solvent-solvent interaction. The difference in the simulation results for the two polymers is all because it is energetically more favorable for the solvent particles in the right compartment to stay closer. After numerous collisions (this is sometimes called entropy-driven), the hairpin conformation emerges as the winner for the polymer on the right.

Figure 5: Higher temperatures.

To make sure that there is no mistake, we ran another simulation in which the two solvents were set to be identically weak-polar. Figure 3 shows that there was no clear formation of a stable conformation for either polymer in a nanosecond-long simulation. Neither polymer curled up.

Next we set the two solvents to be identically strong-polar. Figure 4 shows that the two polymers both ended up in a hairpin conformation in a nanosecond-long simulation.

Another test is to raise the temperature but keep the solvent-solvent interaction in the right compartment three times stronger than that in the left compartment. Can the polymer on the right keep its hairpin conformation when heated? Negative, as shown in Figure 5. This actually is related to denaturation, a process in which a protein loses its stable conformation due to heat (or other external stimuli).

These simulations suggest that our simple-minded model might be able to explain the hydrophobic effect and allow students to explore a variety of variables and concepts that are of fundamental importance in biochemistry. Our next steps are to transfer the modeling work we have done to something students can also do. To accomplish this goal, we will have to figure out how to scaffold the modeling steps to provide some guidance.

Some thoughts and variations of the Gas Frame (a natural user interface for learning gas laws)

A natural user interface (NUI) is the user interface that is based on natural elements or natural actions. Interacting with computer software through a NUI simulates everyday experiences (such as swiping a finger across a touch screen to move a photo in display or just "asking" a computer to do something through voice commands). Because of this resemblance, a NUI is intuitive to use and requires little or no time to learn. NUIs such as touch screen and speech recognition have become commonplace on new computers.

As the sensing capability of computers becomes more powerful and versatile, new types of NUI emerge. The last three years have witnessed the birth and growth of sophisticated 3D motion sensors such as Microsoft Kinect and Leap Motion. These infrared-based sensors are capable of detecting the user's body language within a physical space near a computer with varied degrees of resolution. The rest is how to use the data to create meaningful interactions between the user and a certain piece of computer software.

Think about how STEM education can benefit from this wave of technological innovations. Being scientists, we are especially interested in how these capabilities can be leveraged to improve learning experiences in science education. Thirty years of development, mostly funded by federal agencies such as the National Science Foundation, have produced a wealth of virtual laboratories (aka computational models or simulations) that are currently being used by millions of students. These virtual labs, however, are often criticized for not being physically relevant and not providing hands-on experiences commonly viewed as necessary in practicing science. We now have an opportunity to partially remedy these problems by connecting virtual labs to physical realities through NUIs.

What would a future NUI for a science simulation look like? For example, if you teach physical sciences, you may have seen many versions of gas simulations that allow students to interact with them through some kind of graphical user interface (GUI). What would a NUI for interacting with a gas simulation look like? How would that transform learning? Our Gas Frame provides an example of implementation that may give you something concrete to think about.

Figure 1: The Gas Frame (the default configuration).

In the default implementation (Figure 1), the Gas Frame uses three different kinds of "props" as the natural elements to control three independent variables related to a gas: A warm or cold object to heat or cool the gas, a spring to exert force on a piston that contains the gas, and a syringe to add or remove gas molecules. The reason that I call these objects "props" is because, like in film making, they mostly serve as close simulations to the real things without necessarily performing the real functions (you don't want a prop gun to shoot real bullets, do you?).

The motions of the gas molecules are simulated using a molecular dynamics method and visualized on the computer screen. The volume of the gas is calculated in real time using the molecular dynamics method based on the three physical inputs. In addition to the physical controls through the three props, a set of virtual controls are available on the screen for students to interact with the simulation such as viewing the trajectory path or the kinetic energy of a molecule. These virtual controls support interactions that are impossible in reality (no, we cannot see the trajectory of a single molecule in the air).

The three props can control the gas simulation because a temperature sensor, a force sensor, and a gas pressure sensor are used to detect student interactions with them, respectively. The data from the sensors are then translated into inputs to the gas simulation, creating a virtual response to a real action (e.g., molecules are added or subtracted when the student pushes or pulls a syringe) and a molecular interpretation of the action (e.g., molecules run faster or slower when temperature increases or decreases).

Like in almost all NUIs, the sensors and the data they collect are hidden from students, meaning that students do not need to know that there are sensors involved in their interactions with the gas simulation and they do not need to see the raw data. This is unlike many other activities in which sensors play a central role in inquiry and must be explicitly explained to students (and the data they collected must be visually presented to students, too). There are definitely advantages of using sensors as inquiry tools to teach students how to collect and analyze data. Sometimes we even go extra miles to ask students to use a computer model to make sense of the data (like the simulation fitting idea I blogged before). But that is not the reason why the National Science Foundation funded innovators like us to do.

The NUIs for science simulations that we have developed in our NSF project all use sensors that have been widely used in schools, such as those from Vernier Software and Technology. This makes it possible for teachers to reuse existing sensors to run these NUI apps. This decision to build our NUI technology on existing probeware is essential for our NUI apps to run in a large number of classrooms in the future.

Figure 2: Variation I.

Considering that not all schools have all the types of sensors needed to run the basic version of the Gas Frame app, we have also developed a number of variations that use only one type of sensor in each app.

Figure 2 shows a variation that uses two temperature sensors, each connected to the temperature of the virtual gas in a compartment. The two compartments are separated by a movable piston in the middle. Increasing or decreasing the temperature of the gas in the left or right compartment through heating or cooling the thermal contacts in which the sensors are applied will cause the virtual piston to move accordingly, allowing students to explore the relationships among pressure, temperature, and volume through two thermal interactions in the real world.

Figure 3: Variation II.

Figure 3 shows another variation that uses two gas pressure sensors, each connected to the number of molecules of the virtual gas in a compartment through an attached syringe. Like in Variation I, the two compartment are separated by a movable piston in the middle. Pushing or pulling the real syringes will cause molecules to be added or removed from the virtual compartments, allowing students to explore the relationships among number of molecules, pressure, and volume through two tactile interactions.

If you don't have that many sensors, don't worry -- both variations will still work if only one sensor is available.

I hear you asking: All these sounds fun, but so what? Will students learn more from these? If not, why bother to go through these extra troubles, compared with using an existing GUI version that needs nothing but a computer? I have to confess that I cannot answer this question at this moment. But in the next blog post, I will try to explain our plan for figuring this out.