Difference between revisions of "Team:KU Leuven/Modeling/Hybrid"

When we try to calculate the physical “chemotactic force”, propelling bacteria towards chemoattractants or away from chemorepellents, we find that it is not easily measured nor derived. Therefore, as a workaround we base the equation of motion on (6), a stochastic differential equation (SDE) that describes the motion of a single particle in a N-particle system that is governed by a Keller-Segel type PDE in the limit of $N \rightarrow \infty$. This Keller-Segel type PDE (7) describes the evolution of a bacteria density field $n$ moving towards some chemoattractant $S$.

Put differently, when infinitely many particles obey (6), they will exhibit Keller-Segel type spatial dynamics as in (7). In a sense, we’re using a “reverse-engineered” particle equation that corresponds to the Keller-Segel field equation. A detailed theoretical treatment of (6) is outside the scope of this model description because it contains a stochastic variable. The traditional rules of calculus do not apply anymore for stochastic differential equations and a different mathematical theory called Itô calculus is required. It is sufficient to say that the second term containing $dW$ accounts for Brownian motion in the form of random noise added to the displacement of the agents, causing them to diffuse, and that it is governed by the diffusion coefficient $\mu$.

The first term in (6) on the other hand is easily understood as an advective or drift term (net motion) dependent on S, pushing the agents along a positive gradient (for negative chemotaxis the sign is reversed). The chemotactic drift hence points towards an increasing concentration of the chemoattractant. The advective properties are governed by the chemotactic sensitivity function $\chi (S)$. For our model we used a function of the form (8).

The first important thing to note is that we assume $\chi (S)$ to be proportional to $1/S$. This is because Keller and Segel proved that their corresponding PDE model only yields travelling wave solutions if $\chi (S)$ contains such a singularity at some critical concentration $S_{crit}$, and multiplying by $1/S$ is the simplest way to introduce a singularity at $S_{crit} = 0$. Secondly, the proportionality constant is composed of two factors, namely the bacterial diffusion coefficient $\mu$ and chemotactic sensitivity constant $\kappa$. This is done for two reasons. Firstly, when $\mu$ is lowered, both chemotactic and random motion is reduced, which emulates the state of inactivated motility due to high or low concentrations of AHL. Secondly, defining a separate chemotactic sensitivity constant allows us to examine the effect of the relative strength of chemotaxis versus random motion on pattern formation. Conceptually, this term can be viewed as a sort of chemotactic force, defined with regards to a mobility coefficient $\mu$ instead of a frictional coeffient, which are the inverse of each other.

In addition to chemotaxis and diffusion, cell-cell interactions play an important role in pattern formation and also need to be modeled. Bacteria have finite size and therefore multiple bacteria cannot occupy the same space. Moreover, an important mechanism in our system is the aggregation of cells A due to the sticky adhesin protein membrane. To take these mechanisms into account we modeled two types of cell-cell interactions: the purely repulsive interaction of cell B with another cell B and with cell A, and the attractive-repulsive interaction of cell A with another cell A. The interaction between two cells is usually expressed by a potential energy curve defined over the distance between the centers of mass of the two cells (Figure 2).

Figure 2

Cell-cell interaction potential. Click to enlarge.

Note that the potential energy remains constant after a certain distance, which means that the cells stop interacting. Also, as two cells move closer together, they hit a wall where the potential energy curve abruptly goes to infinity. The reason for this is that two cells cannot occupy the same space and therefore smaller intercellular distances are not allowed. Implementing this theoretical potential is however not possible because the displacement of bacteria is a stochastic process and the bacteria could randomly jump beyond the potential wall, where the force is ill defined. Therefore, we’ve decided to define a piecewise quadratic potential (9), which results in a piecewise linear force that resembles Hooke’s law, but with three different “spring constants” acting in different intervals of intercellular distances (10), as illustrated in Figure 3. The force is defined with respect to the unit vector pointing towards the other cell, meaning that a positive force corresponds to an attractive force and vice-versa.

Figure 3

Cell-cell interaction potential and force curves. Click to enlarge.

Between A type cells, there is a region of attraction $(R_0 \leq r_{ij} < R_{cutoff})$, where the force points towards the other cell, hence moving them closer together. In the repulsive domain $(r_{ij} < R_0)$, two regions were defined, emulating lower repulsive forces $(\frac{R_0}{2} \leq r_{ij} < R_0)$ and higher repulsive forces due to a higher spring constant when the cells are even closer $(r_{ij} < \frac{R_0}{2})$. For the purely repulsive interaction scheme there is no attraction and therefore the spring constant for $R_0 \leq r_{ij}$ is zero. More details about the implementation of the cell-cell interaction scheme, more specifically regarding the nearest-neighbor search algorithm, can be found in the paragraph on the agent-based module below.

Now we are ready to construct the equation of motion for cell type A and B as a superposition of an adapted Keller-Segel SDE (6) and cell-cell interaction forces (10), yielding (11).

Bacteria of type A are not attracted nor repelled by leucine, so the chemotactic term falls away. All cell-cell forces are summed up to find a net force, taking into account the two different potentials due to the different interaction types. As discussed before, this net force times a constant yields the velocity due to that force, which is then multiplied by $dt$ to obtain the displacement. For B type cells, the chemotactic term models the repulsive chemotaxis away from leucine. The chemotactic sensitivity function (12) has a negative sign signifying that B type cells are repelled by leucine. The cell-cell interaction term in this case is simpler because B type cells only interact repulsively.

Note that the diffusion coefficient of cell types A and B (13) switches based on the local concentration of AHL relative to a threshold AHL value, which simulates the dependency of cellular motility on AHL. For B type cells the cellular motility depends explicitly on AHL due to the synthetic genetic circuit we have built into them. On the other hand, in our model the motility of A type cells should not depend on AHL directly, but since high concentrations of AHL are caused by high densities of aggregated A type cells, the bacteria typically will not be motile in high AHL concentrations because they stick to neighboring cells.

The agent-based module is now fully defined but one crucial issue was skipped: AHL and leucine concentrations are calculated using PDEs and are therefore only known at grid points. Agents on the other hand reside in the space between grid points and require local concentrations as inputs to calculate their next step. This problem is part of the coupling aspect in our hybrid modeling framework and is discussed below.

As described above, the agents’ effect in the PDE is modeled as a source term that is proportional to the agent density. This approach is essentially the same approach taken in the colony level model for the bacterial production of AHL and leucine. However, in the colony level model the bacteria density is explicitly calculated at the grid points, while the agent-based model essentially considers a set of points scattered in space. A simple first-order approach would be to determine the closest grid point to any agent and simply increment a counter belonging to that grid point. This results in a histogram, which can be used directly to represent the agent density. However, the resulting density is a blocky, nonsmooth function which poorly represents the underlying agent distribution. The effect of a single agent is artificially confined to a single grid point, while in reality an agent’s influence could reach much further than a single grid point. The shape of a histogram is also very dependent on the bin size, which in this case corresponds to the grid spacing so it cannot be independently tuned. To decouple grid spacing and agent density and achieve a smoother density function, we made use of a more sophisticated technique called kernel density estimation (KDE).

KDE is used in statistics to estimate the probability density of a set discrete data derived from a random process. The basic idea consists of defining a kernel function that represents the density of a single data point, then centering kernel functions on every data point and summing them all up to achieve a smooth overall density function, as demonstrated in the Figure 4.

Figure 4

Kernel density estimation. Click to enlarge.

This kernel function can be anything as long is it continuous, symmetric and integrates to 1, since it represents one data point or one agent in our case. Some of the most common kernel functions include Gaussian kernels, triangular kernels and Epanechnikov kernels (14), which are shown in Figure 5.

Figure 5

Gaussian, triangular and Epanechnikov kernel functions. Click to enlarge.

In practice, scaled versions of standard kernel functions are used, which are of the form (15).

Importantly, the scaled functions inherit the kernel function properties, but are either broader or narrower. The degree to which the shape of a kernel function is stretched or squeezed depends on the scaling factor h, which is why it is called the bandwidth, see Figure 6.

Figure 6

Epanechnikov kernels using different bandwidths. Click to enlarge.

This parameter gives us the freedom to define how far the influence of an agent reaches and how smooth the resulting density function looks like. An example is given in Figure 7, where 100 agents were sampled from a normal distribution. The red dotted line indicates the true underlying distribution, while the blue line is the kernel density estimation calculated using different bandwidths. The left graph shows a undersmoothed (small bandwidth) KDE, which oscillates wildly. The right graph on the other hand is oversmoothed (large bandwidth), leading to a misleadingly wide KDE.

Figure 7

Kernel density estimation using different bandwidths. Click to enlarge.

In summary, using kernel density estimation we can express the agent density in the form of (16), providing the appropriate input for the PDE model.

The extension to higher dimensions is called multivariate kernel density estimation and is rather non-trivial since the bandwidth is then defined as a matrix instead of a scalar, which allows for many more variations on the same basis kernel function. A detailed analysis of multivariate kernel density estimation will not be given here. It suffices to say that for our 2-D hybrid model we used the simplest possible bandwidth matrix, which is the identity matrix times a constant.

The final component of our hybrid model is the mapping of the PDE model to the agent-based model. The latter model works with discrete objects that have continuous coordinates, meaning that they can be located at any point of the domain. As we have seen in (11), the agents need the local concentration of AHL and leucine, as well as the gradient of leucine in order to update their positions. In the PDE model however, the domain is discretized into a grid and concentrations are only defined at grid points. Therefore, in order to transfer information from the PDE model to the agent-based model we need to translate these grid values into values for any given position within the domain. We achieved this for the 1-D hybrid model by taking the two nearest grid points to any agent and employing linear interpolation to derive an approximate local field value. Similarly, for the 2-D hybrid model we took the four nearest grid points and employed bilinear interpolation, as explained in box 3. With the end result (B3.3), the concentration or gradient of AHL or leucine can be derived at any point of the domain. Therefore, the agent-based module has access to the information that it needs from the PDE module.

The cell-cell interactions have already been fully described in the paragraphs above. However, solving the equation of motion of an agent in its current form requires the computation of the force due to every other agent. If we take N to be the number of agents, that means $N*(N-1)$ amount of force computations are needed in total and therefore the computation time grows as $O(N^2)$. To put this in perspective, if we simulate a thousand agents, the amount of interactions adds up to around one million. This puts a heavy computational load on the model which can be reduced greatly by using a smarter algorithm. The crucial observation here is that the force goes to zero when the distance between two cells is larger than $2\cdot r_{cutoff}$. An agent therefore only interacts with its nearest neighbors, meaning that the naive implementation wastes a lot of time computing interactions which have no effect anyways. Moreover, as the simulation progresses, an agent’s neighbors do not vary greatly from one timestep to another. A more efficient approach then consists of periodically searching for the nearest neighbors within a fixed radius, storing them in a list for every agent and updating their coordinates for all intermediate timesteps.

Since the agents are programmed to repel each other when they approach each other too closely, they will evolve to a rather uniform distribution. The most appropriate fixed-radius nearest neighbor search algorithm in that case is the cell technique. In this algorithm, the agents are structured by placing a rectangular grid of cells over the domain and assigning every agent to a cell (fig9). Determining which cell an agent belongs to is easily done by rounding down the x and y-coordinates to the nearest multiple of the grid spacing. If the grid spacing is chosen so that interactions do not reach further than adjacent cells, every agent only needs to look for neighbors in 9 cells (its own cells plus 8 adjacent cells) instead of the entire domain.

In contrast to PDE models, there is no danger of instability due to a large timestep in agent-based models. However, since the time step determines how far a cell can “jump” away from its previous position, we need to ensure that it is not so large as to negate the effect of intercellular interactions. To understand this, we first discretize the Brownian motion term in the agent’s equation of motion (17).

The details of this discretization scheme goes beyond the scope of this text and will not be given here. It can be proven that this term follows a distribution as (18).

Now assume that there are two A type cells who are exactly $2 \cdot r_0$ apart. If the timestep is too large, they might “jump” out of each other’s influence at the next iteration. This situation is highly unrealistic because the cells actually should feel each other’s pull while they randomly drift away from each other. If the timestep is too large, there’s a high probability that the cells detach without experiencing the potential at all and therefore the probability of detachment is artificially inflated. Of course, we can never completely eliminate the possibility of a sudden detachment because the random is sampled by a normal distribution, but we can define a desired probability of sudden detachment $P^-$. Then, for a given $r_0$, $r_{cutoff}$ and $\mu$, we can determine how small the timestep has to be for the sudden detachment to happen only with probability $P^-$. First, we assume two A type cells with coordinates $(x_1,y_1)$ and $(x_2,y_2)$ that are at a distance $2 \cdot r_0 = R_0$ from each other. We then define the coordinates at the next timestep as stochastic variables $X_1$, $Y_1$, $X_2$ and $Y_2$, which are distributed as (19).

We are not interested in the future coordinates as such, but rather in the future intercellular distance, which is composed of these future coordinates and therefore also is a stochastic variable. For ease of calculation, we work with the square of this variable, which we name as R² and define as (20).

It is easy to see that $X_1-X_2$ and $Y_1-Y_2$ are both noncentral (mean is not equal to zero) normally distributed variables as well. Therefore $R^2$ obeys a noncentral chi-squared distribution with 2 degrees of freedom. Before proceeding further, we have to make our $R^2$ conform to a standard noncentral chi-squared distribution. First we normalize $X_1-X_2$ and $Y_1-Y_2$. We observe that $X_1-X_2$ and $Y_1-Y_2$ are distributed as (21).

Secondly, because the means of the $X_1-X_2$ and $Y_1-Y_2$ are not equal to zero, we need to account for this by calculating the noncentrality parameter $\lambda$. This is equal to the sum of squares of the means of the normalized $X_1-X_2$ and $Y_1-Y_2$ variables, which of course equals (22), corresponding to a scaled version of the original intercellular distance.

We then define the standardized variable $\bar{R}^2$ as (23).

To recap, the meaning of $\bar{R}^2$ is the scaled squared intercellular distance after updating once with timestep $\Delta t$ starting from an initial distance of $r_0$. Then we define $F$ as the cumulative distribution function for the random variable $\bar{R}^2$ as (24).

To then finally calculate appropriate timestep for a given $r_0$, $r_{cutoff}$, $\mu$ and desired $P^-$ we take the inverse cumulative distribution function and equate it to the intercellular cutoff distance, after appropriate scaling and squaring (25).

Note that both the left hand side and right hand side both depend on the $\Delta t$ in a nonlinear way, therefore a nonlinear solver is required to solve this equation. As stated previously, it is not possible to completely avoid that the cells will jump out of each other's influence in a single timestep. However, equation (25) allows us to calculate the timestep $\Delta t$ that will result in cells detaching in a single timestep with a probability $P^-$ that we can specify.

As mentioned earlier the concentrations of AHL and Leucine are modeled using partial differential equations. In the colony level model these equations are solved explicitly. Explicit schemes do not require a lot of work per time step, but unfortunately are not unconditionally stable. In two dimensions the grid ratios $dt/dx^2$ and $dt/dy^2$ can not exceed $dt/dx^2 + dt/dy^2 \leq \frac{1}{2}$ for the solver to be stable. When computing the solution of the hybrid model this requrement forces us to spend a lot of CPU time solving partial differential equations that could be better spent simulation the agents. Therefore an implicit ADI Alternating direction implicit scheme has been implemented. ADI-schemes are unconditionally stable, which allows it to take large time steps with the PDE solver. We used the following scheme: $$ (1 - \frac{1}{2} \mu_x \delta_x^2) U^{n+\frac{1}{2}} + \frac{1}{4}kU^{n+\frac{1}{2}} = (1 + \frac{1}{2} \mu_y \delta_y^2) U^n - \frac{1}{4}kU^n + \frac{\alpha}{2} \rho_A $$ $$ (1 - \frac{1}{2} \mu_y \delta_y^2) U^{n+1} + \frac{1}{4}kU^{n+1} = (1 + \frac{1}{2} \mu_x \delta_x^2)U^{n+\frac{1}{2}} - \frac{1}{4}kU^{n+\frac{1}{2}} + \frac{\alpha}{2} \rho_A $$ In the equations above $\mu$ denotes grid ratios and $\delta^2$ central differences. The production and degradation terms have been incorporated at every time level with a factor of $\frac{1}{4}$. The image below shows the computational molecule of the ADI scheme we chose to implement:

Figure 4

ADI-Molecule. Click to enlarge.

Text about boundary conditions.

In order to benefit from the implicit PDE-solver described above the agent's time steps are chosen smaller then the time steps of the PDE solver. However type A cells produce molecules continuously as they move trough space. Therefore we record their positions and average over the kernel functions centered at past positions since the last PDE evaluation. That way we avoid blurring the results of the PDE solver too much, when the time step of the agents is reduced. Last but not least we want to point out the relationship between kernel bandwidth and the grid on which the PDE is solved. The larger the kernel bandwidth $h$ is chosen the coarser the PDE grid can be without loosing cells between grid points. If the the diameter of the kernel functions is smaller then the distance between PDE grid points it can happen, that the kernel does not overlap with any PDE grid points. In this case a cells contribution to the molecule concentrations at the next time step could be lost. When the PDE grid is widened to save computing time it is thus necessary to increase the Kernel bandwidth as well. However this increase can lead to a situation where the Kernel function covers significantly more space then the actual diameter of a bacterium. In these cases the Kernel function can be interpreted as a probability function for a cells position. However we avoided too large bandwiths and PDE grids by running our 2-D simulations at the Flemish Supercomputer Center (VSC).

Figure 6

Logo of the Flemish Supercomputer Center. Click to enlarge.

Text about bandwidth tuning.