Difference between revisions of "Team:KU Leuven/Modeling/Hybrid"
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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 $\mu A$, 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 $2*r_0$ = $R_0$ 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 (eq. 17). 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 $R^2$ and define as (eq. 18). 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 (eq. 19). 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 (eq. 20), | 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 $\mu A$, 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 $2*r_0$ = $R_0$ 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 (eq. 17). 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 $R^2$ and define as (eq. 18). 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 (eq. 19). 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 (eq. 20), | ||
− | $$ \lambda = ( \frac{x_1 - x_2}{2\sqrt{\mu \triangle t})^2 + (\frac{y_1 - y_2}{2 \sqrt{2 \mu \triangle t})^2}$$ | + | $$ \lambda = ( \frac{x_1 - x_2}{2\sqrt{\mu \triangle t}})^2 + (\frac{y_1 - y_2}{2 \sqrt{2 \mu \triangle t}})^2}$$ |
$$ = \frac{R_0^2}{4\mu \triangle t} \;\;\; (20) $$. | $$ = \frac{R_0^2}{4\mu \triangle t} \;\;\; (20) $$. | ||
corresponding to a scaled version of the original intercellular distance. We then define the standardized variable $R^2$ as (eq. 21). | corresponding to a scaled version of the original intercellular distance. We then define the standardized variable $R^2$ as (eq. 21). |
Revision as of 03:22, 19 September 2015
The Hybrid Model
The hybrid model represents an intermediate level of detail in between the colony level model and the internal model. Bacteria are treated as individual agents that behave according to the Keller-Segel type discretized stochastic differential equations, while chemical species are modeled using partial differential equations.
Model Description
Implementation
1-D Hybrid Model
The video box above shows one dimensional simulation results for the hybrid model. A constant speed and random step simulation has been computed. We observe that the bacteria form a traveling wave in both cases, which is essential for pattern formation. These results are also similar to what we get from the continuous model, which confirms our results.
2-D Hybrid Model
The videos above show simulation videos computed at the Flemish supercomputing center, for three different initial conditions similar to the ones we used for the colony level model. The first and second condition start from 9 mixed or 5 colonies of both cell types, arranged in a block or star shape. These first two gradually separate in a manner similar to what we would we also saw in the colony level model. The result for random initial data is fundamentally different. As the agent based approach allows for better implementation of adhesion large cell type A bands form. The AHL and Leucine produced by the type A bacteria causes the B type cells to move away leading to a pattern which we could not produce using PDEs alone, this beautifully illustrates the added value of hybrid modeling.
Incorporation of internal model
Up until now, we have largely ignored the inner life of the bacteria. This inner life consists of transcriptional networks and protein kinetics. Instead we assumed that AHL and leucine production is directly proportional to the density of type A cells. This only works in theory, since bacteria will be affected by their surroundings and the way their dynamics react to it. For example bacteria surrounded by a large concentration of AHL, will have more CheZ and will react more on the presence of Leucine. Also bacteria have different histories and will have different levels of transcription factors and different levels of proteins in their plasma. The proteins are not directly degraded and will still be present in the cytoplasm of the bacteria long after the network has been deactivated. From this, it is clear that 2 bacteria, although surrounded by the same AHL and leucine concentrations, can show different behavior and reaction kinetics.
This results in a heterogeneity of the bacterial population that has not yet been accounted for. To make up for this anomaly, we decided to add an internal model to every agent. This way we will get more realistic simulations. Every agent will get their own levels of CheZ, LuxR, LuxI and so on and will have individual reactions on their surroundings. We hope that this way we can get closer to the behavior of real bacteria.
References
[1] | Benjamin Franz and Radek Erban. Hybrid modelling of individual movement and collective behaviour. Lecture Notes in Mathematics, 2071:129-157, 2013. [ .pdf ] |
[2] | Zaiyi Guo, Peter M A Sloot, and Joc Cing Tay. A hybrid agent-based approach for modeling microbiological systems. Journal of Theoretical Biology, 255(2):163-175, 2008. [ DOI ] |
[3] | E F Keller and L A Segel. Traveling bands of chemotactic bacteria: a theoretical analysis. Journal of theoretical biology, 30(2):235-248, 1971. [ DOI ] |
[4] | E. M. Purcell. Life at low Reynolds number, 1977. [ DOI ] |
[5] | Angela Stevens. The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems, 2000. [ DOI ] |
Equations
Contact
Address: Celestijnenlaan 200G room 00.08 - 3001 Heverlee
Telephone: +32(0)16 32 73 19
Email: igem@chem.kuleuven.be