Microglia

      The initial microglia count dictates the number of microglial agents initially in the simulation. The microglia concentration designates how many microglia are represented per agent. Thus, if the intial count is 40 and the concentration is 10, then 400 microglia are represented in the simulation. By using the concentration parameter one can change the number of microglia in the simulation without changing the computation power. The microglial agent is placed (or centered) within a grid space. The number of agents centered in any one grid space cannot exceed the maximum density parameter.

      At initialization, a random pairing of (x,y) is generated. If there is room (meaning that the maximum density parameter will not be exceeded) to place a microglial agent in this grid space, a microlgial agent will be placed in the grid space. If there is no room, a new random pairing of (x,y) is generated and tested. This process continues until a number of microglial agents equal to the initial microglia count are in place.

Absorption of Soluble Amyloid Protein

      Microglia absorb soluble amyloid according to the Michaelis-Menten kinetics discussed above. The maximum sAB microglia uptake rate is k and the half max amyloid binding conc is h. Because this occurs on the macro time scale, the amount of soluble amyloid changed in the same grid space as the microglial agent, call it DS, is

DS = q*DT*k*S/(h+S)
where q is the microglia concentration and S is the concentration of soluble amyloid in the same grid space as the microglial agent. DS may be modified if the cell is nearing its capacity for soluble amyloid defined by the fatal sol-AB dosage parameter. If DS is great enough so that if it were added to the stored soluble amyloid in the microglium, the storage concentration would exceed the product of the capacity and density, then DS is decreased to the maximum concentration that can be stored without exceeding capacity. Having determined DS, the concentration of soluble amyloid in the same grid space as the microglial agent is decreased by DS and the concentration of soluble amyloid within the agent is increased by DS.

Neutralization of Stored Soluble Amyloid Protein

      In addition to absorbing soluble amyloid, it is assumed that microglia can neutralize the concentration of soluble amyloid that it has stored. It does this at a constant rate defined by the sol-AB neutralization rate parameter. In this case, the change in concentration within the cell is simply the product of the microglia concentration, the neutralization rate and the macro time step. Of course, care is taken so that the concentration of soluble amyloid in storage is never negative.

Secretion of IL-1B

      Microglia secrete IL-1B. For details on how this is done, please see IL-1B under Chemicals.

Movement

      The motility time delay parameter determines how many macro time steps a microglial agent must wait before moving. If there is no delay, then the speed of movement is simply the length of a grid space, DX, divided by the macro time step, DT. In general, if m is the number of steps between movement, then the corresponding speed is v = DX/(DT+1).

      Each microglial agent has a counter which gets decremented every time step. When the counter reaches zero the cell attempts to move according to the rules specified below. Also at this time, the counter gets reset to the motility time delay value. When microglial cells are initialized, the counter takes any integer value between zero and the time delay, uniformly distributed. This allows for movement to be asynchronous, meaning that not all microglia will move at the exact same time.

      Movement of microglia can be hampered by the presence of amyloid fibers. If the concentration of fibers in the grid space of the microglial cell is F and the concentration of fibers at which a cell has a 50-50 chance of being stuck is called G (this is the half-sticking fiber level parameter), then the the probability, P, that a cell gets stuck is

P = F/(F + G).
This is tested in a Monte Carlo fashion meaning that a uniformly distributed random variable between 0 and 1 is generated and if its value is less than P, then the cell is stuck contingent on the concentration of soluble amyloid present. If the concentration of soluble amyloid, S, is great enough, then normally stuck cells are free to move depending on the concentration of soluble amyloid at which a cell has a 50-50 chance of being unstuck (call this H which is the half-sticking sAB level parameter). The probability, P, of a cell to remain stuck is
.
This is tested in the same Monte Carlo fashion.

      Microglia move chemotactically up the soluble amyloid gradient. Thus, the first step in cell movement is to determine which of the surrounding grid spaces has the greatest concentration of soluble amyloid. The grid spaces sampled can be labeled as 

	0 1 2
	3 4 5
	6 7 8
where 4 is the current grid space of the microglial agent. In the case where the cell is on a boundary, periodic boundary conditions are assumed. After sampling the immediate neighborhood, the direction of greatest concentration is noted. The cell will move in this direction in a Monte Carlo fashion with the probability of winning given by the chemotactic sensitivity parameter. (Simply put, the cell will move in the direction of highest concentration with a probability equal to the chemotactic sensitivity parameter.) If the designated direction loses or all concentrations are the same so that there are no designated directions, then a direction 0-8 is chosen at random, uniformly distributed. Having finally decided on a direction of movement, the cell must check to see if room is available for movement. Room exists if no astrocytes are in the grid space and the addition of another microglial agent will not make the number of microglial agents in the grid space exceed the maximum density parameter. If there is no room, then the microglial agent will stay in its current grid space.

Cell Death

      If cell death is turned on, then one of two factors will determine whether a microglial agent will die. If the concentration of absorbed soluble amyloid ever reaches the fatal sol-AB dosage parameter, then the cell will die. A cell could also die based on old age. We assume that the probability that a cell dies by age T is based on a sigmoidal distribution,

where A is the typical cell duration. Solving the above equation for T allows us to determine the time of death based on the cumulative distribution function
.
By generating random values of P between 0 and 1, uniformly distributed, we can determine a corresponding random age of death. Thus, when the cell is created (or initialized), the time of death is known and this value is simply placed in a death counter (as T/DT) that gets decremented every macro time step. If the fatal soluble amyloid dosage is never reached before the death counter reaches zero, then death takes place when the counter reaches zero.

Mitosis (Cell Division)

      If cell mitosis is turned on, then four criteria determine whether mitosis can occur. These factors are the length of cell cycle, half-sticking fiber level, poison effects on mitosis, and chance of mitosis. All four criteria must be satisfied in order for mitosis to occur.

      The length of cell cycle determines the time at which mitosis can occur. Every microglial agent has a counter which has the cell's current age (in number of macro time steps since the cell was initialized). The length of cell cycle, call it C, is converted into number of macro time steps, M = C/DT. Whenever, the age counter is a multiple of M, mitosis can occur. Of course, this process can be changed, so that all microglial agents do not divide at the same time as follows: The length of the cell cycle can determine the mean of a normal distribution with the standard deviation another parameter to be added to the simulation. The timing for mitosis can then be handled much like cell death above, where each time a microglial agent is initiated, a time for cell mitosis can be randomly generated based on the normal distribution (or any other distribution). Cells can then keep track of a mitosis counter which gets decremented every macro time step. When the counter reaches zero, the other criteria are checked and the counter gets reset (stochastically based on the mitosis distribution like it was at initiation).

      Because amyloid fibers are deemed hazardous to microglia, presence of fibers may prevent mitosis. In our current formulation, if the concentration of amyloid fibers in the same grid space as the microglial agent is greater than the half-sticking fiber level, then mitosis is prevented. If the concentration of amyloid fibers is less than the half-sticking fiber level, then the other criteria are checked.

      The amount of soluble amyloid within the microglia (having been absorbed) also affects mitosis. The poison effects on mitosis parameter accounts for this. If the parameter equals one, then the cell is deemed poisoned if any soluble amyloid is stored inside the cell. If the parameter equals zero, then soluble amyloid does not act as a poison which prohibits mitosis. Values in between dictate a percentage of the capacity (the fatal sol-AB dosage parameter) at which the cell is poisoned. In this case, let E be the value of the poison effects parameter, S be the concentration of soluble amyloid within the cell, q be the microglia concentration and C be the capacity of soluble amyloid within a cell. The microglial agent is poisoned if S is greater than q(1-E)C in which case mitosis cannot occur. Otherwise, the other criteria are checked.

      In our current formulation, all the other criteria are deterministic (the condtions are either met or not). To add a stochastic component to the model, the chance of mitosis parameter is checked in a Monte Carlo fashion. Thus, if all the other criteria have been met, a uniformly distributed random variable between 0 and 1 is generated. If this number is less than the chance of mitosis parameter, mitosis occurs. Otherwise, nothing happens until the next time mitosis can be checked.

      If all the conditions for mitosis are met, then the program will attempt to add another microglial agent to the simulation. It first checks to make sure that there is enough memory (currently, the simulation can hold 1000 microglial agents). A new microglial agent is then initiated in the same grid space as the old one. The concentration of soluble amyloid in storage in the old agent is divided so that the old agent holds half and the new agent begins with half. If the addition of the new microglial agent would make the number of microglial agents exceed the maximum density parameter, then the new agent simply replaces the old one.


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