Synergy/Particle model
Introduction
One big concern in our project was the efficiency of propane production. To solve this problem we wanted to use micelles to hold enzymes together and speed up the reactions. By having two of our most inefficient enzymes close together we try to increase the rate of the reactions. Our only question regarding this is: Does it actually work?
To address this not so trivial question, the modeling team assembled and thought about the problem. This problem couldn’t be described easily with simple differential equations since then notions of distance, proximity and their relation with enzyme reaction rates would have to be thoroughly researched as well.
Instead, the modeling team wondered: Is it possible to model the enzyme reactions like they happen in a cell? In the real world, there are no simple numbers inside the cell: Notions like reaction rate and enzyme kinetics arise from the chaotic fluctuations between molecules inside the cell. Enzyme reactions between enzymes and substrates are dependent on many factors, such as molecules having the right energy and orientation.
As such, this problem is quite hard and would undoubtedly need a lot of research. Instead, we decided to simplify the situation: Could we simulate enzymes and substrates as particles in a cell and model enzyme reactions by simulating their interactions with each other?
The model
To compare between enzymes being close together and not being close together, the model simulates enzyme reactions inside a cell. The model behaves stochastically, and needs to be ran multiple times to ensure reliable results, as its results will vary according to pseudorandom variables.
The model is a computer program made with Python that simulates a space filled with enzymes and substrates which react with each other. A simplified flowchart of the program is presented below.
Now, let’s go through the different phases of the simulation:
Initialization: The program loads a settings file filled with information concerning the simulation, and creates a simulation according these specifications. With this file, the user can, for example, specify the length of the simulation, the different substrates with their amounts and masses, as well as the different enzymes and the types of substrates and products they either consume or produce.
Particle movement: The model moves particles according to Brownian motion in water. With particles of this size the governing attribute these particles have is their radius.
The movement of particles under the influence of brownian motion follows a normal distribution. According to ____, the mean squared displacement of particles experiencing brownian motion is proportional to the time interval: \[ \left( | r(t + dt) - r(t) |^2 \right) = 2 \cdot d \cdot D \cdot dt \], where r(t) is the position of the particle, d is the number of dimensions, D is the diffusion coefficient and dt is time interval. For us to generate correct brownian motion for our particles, we need to scale the normal distribution with a factor \[ k = \sqrt{D \cdot d \cdot dt} \]. For our simulation, d=2 and dt is a time interval defined by the user. D or the diffusion coefficient is calculated from the Einstein relation: \[ D = µk_BT \] where \(k_B\) is the Boltzmann’s constant and T is the temperature, and \( µ \) is the particle’s mobility: \[ µ = \frac{1}{6 \cdot \pi \cdot \eta \cdot r} \] where \( \eta \) is the dynamic viscosity of the fluid and r is the particle’s radius.
Getting the radius of the particles is a bit trickier. To calculate this, we use the same method as in our Micelle modeling. Based on the information in this paper we can calculate the enzymes’ radii from their mass.
Proteins' approximate density is 1.37 g/ml. Thus, the specific partial volume (or inverse density) is 0.73 ml/g. From this, we can calculate that the volume of an enzyme mainly consisting of amino acids is \[V(nm^3)=\frac{0.73\tfrac{cm^3}{g}10^{21}\tfrac{nm^3}{cm^3}}{6.023\cdot 10^{23} \tfrac{Da}{g}} M(Da).\] Then, if we approximate the enzymes as spheres with a volume \( V(r) = \frac{4}{3} \cdot \pi \cdot r³ \), we can calculate the enzymes' radii:\[R_{min(nm)} = \left( \frac{3V}{4\pi} \right)^{1/3}\ = \left( \frac{3\cdot \frac{0.73\tfrac{cm^3}{g}10^{21}\tfrac{nm^3}{cm^3}}{6.023\cdot 10^{23} \tfrac{Da}{g}}}{4\pi} \right)^{1/3} \cdot \left( M(Da) \right) ^{1/3}
= 0.066 \cdot (M(Da))^{1/3} \] For substrates this is not so easy. To simplify the situation we used the same method for calculating the approximate radii for the substrates as well, based on their molar mass.
Reactions: The simulation considers that a reaction is possible only when a right type of substrate and a right type of enzyme are close enough to react. When this happens, the simulation randomly decides if the reaction actually took place with user-defined probability. If the reaction happened, the simulation changes the substrate to the enzyme’s product and makes both the substrate and enzyme unable to react for a short amount of time.
Data logging: the program collects only one type of data, and that is the particle numbers of each substrate at each point in time that the simulation runs. This makes it possible to make figures of the reactions and to determine the different reaction rates.
Post-Simulation tasks:
After the simulation, the program creates a data file from the simulation data it gathered. This data file has data about the amounts of substrates at each point in time.
Assumptions and other background
The model is built on a big list of assumptions. Some of these assumptions were made to simplify the situation, some of them were made to include some aspects of enzyme activity. Here are some of the most important ones.
Fundamental assumptions that are not changeable in the simulation
Enzymes and substrates are smallish particles floating in the cell. These particles move because of brownian motion. This motion is affected by the size of the particles.
Enzymes and substrates only react when they are in close enough proximity with each other. In addition to this, enzymes and substrates only react with the correct type of substrate and enzyme, respectively.
Reaction between enzymes and substrates are difficult to model. Since we don’t have a good way of predicting when a reaction should happen given that an enzyme and a substrate meet, we decide this by giving a reaction a probability to succeed and polling this chance every time a substrate and enzyme meet.
Enzyme concentration and therefore particle number stays constant during the simulation.
Substrate amounts are either decided in the beginning of the simulation or they are affected by the enzyme reactions.
The model keeps the enzymes and substrates on a 2D plane, as opposed to a 3D space. This is because we assume that the model does not change much between 2D and 3D space.
To simplify this model, enzymatic reactions do not take into account cofactors or multiple substrates.
Optional assumptions that the user can either activate or leave unused
Enzyme reactions are not instantaneous, instead they take some time that is specific to the enzyme.
Some enzymes may be in close proximity with each other, if they are e.g. fused together or in micelles.
Some or all of the substrates can have a constant concentration, even if some enzymes consume them. This is the case if the substrate amount (not necessarily the concentration) is so huge when compared to enzyme amount that the enzyme’s effect on its amount is negligible.
Results
Our awesome results!
Implications of the model
Here talk about what the results actually say, containing restrictions
