Systems

A system is a group of parts that are interconnected and affect each other

• The parts are separated and well defined
• Instead of parts, we can say items or components
• They have interactions, for example a process that transforms some items into others

Example: cell-food system

How to draw the system

The arrows going into a box show which items are required for the process

• The items that point towards a process are the inputs that will be consumed by the process
  • When a item is consumed by a process, its amount gets smaller
• The arrows going out of a box show which are the products of the process
  • The amount of these items will increase with the process.

Each item has two values

At any given moment, there is some amount of each item in the system

• for example, the number of cells

Also, we can know how much did the amount changed between the previous moment and this moment

• for example, the growth of cells

Representing items in R

Each item (circle) has two values, that change with time

We will represent these values with vectors

• The amount of item will be called item
• The change of item will be called d_item
  • We read it “delta-item”

These are dynamical systems

Dynamical system means that the system changes with time

We will represent time by a positive integer number

Time i represents “now”. The units can be seconds, hours, days, years, or whatever is best to understand the system

If we use “days”, then i-1 means “yesterday”. It can also be “last year” or “one second ago”

Systems are in a state

1. The particular condition that someone or something is in at a specific time

   • water in a liquid state.

2. The amounts of all items (circles) in a system.

   • cells[i]

   • food[i]

The state of the system changes with time

In any fixed time, the state is all the values of items’ quantities

• one value for each item
• Examples:
  • cells[i], food[i]
  • Concentration of H, O, Water
  • Number of foxes and rabbits
  • Number of primers and DNA molecules

Finding the formulas for boxes

For each box in the graph we get only one term

We multiply the rate constant and each of the amount variables of the circles connected by incoming arrows

We use the index i-1. Processes depend on the previous values

If there are several incoming arrows from the same circle, then the variable is multiplied several times

The outgoing arrows are not important in this part

If there are several arrows

In the last slide we had several arrows between the cells circle and the eating box

It is easy to see that we only care about the resulting number and direction of arrows

Two input - One output = One input

Initial conditions

The last missing piece are the initial values of the circles’ variables.

The value of cells[i] depends on the value of cells[i-1], and we can only calculate that for i >= 2

We will use the variables cells_ini and food_ini to define the values used in cells[1] and food[1]

For the delta variables, we can assume that they are initially zero.

All code togethers

N <- 168
cells   <- d_cells <- rep(NA, N)
food    <- d_food  <- rep(NA, N)
cells[1] <- 1
food[1]  <- 20
d_cells[1] <- d_food[1]  <- 0

for(i in 2:N) {
  d_cells[i] <- cells[i-1]*food[i-1]*eating_rate
  d_food[i]  <- -cells[i-1]*food[i-1]*eating_rate
  cells[i]  <- cells[i-1] + d_cells[i]
  food[i]   <- food[i-1]  + d_food[i]
}

We will write it as a function

cell_culture <- function(N, eating_rate, cells_ini, food_ini) {
  cells   <- d_cells <- rep(NA, N)
  food    <- d_food  <- rep(NA, N)
  cells[1] <- cells_ini
  food[1]  <- food_ini
  d_cells[1] <- d_food[1]  <- 0
  for(i in 2:N) {
    d_cells[i] <- cells[i-1]*food[i-1]*eating_rate
    d_food[i]  <- -cells[i-1]*food[i-1]*eating_rate
    cells[i]  <- cells[i-1] + d_cells[i]
    food[i]   <- food[i-1]  + d_food[i]
  }
  return(data.frame(cells, food, d_cells, d_food))
}

This is what you have to remember

system_name <- function(N, rates, initial_cond) {
  state   <- d_state <- rep(NA, N)
  state[1] <- initial_cond
  d_state[1]  <- 0
  for(i in 2:N) {
    d_state[i] <- some formula
    state[i]  <- state[i-1] + d_state[i]
  }
  return(data.frame(state, d_state))
}

To know more …

This online material will show you the big picture

• Systems theory in Wikipedia
• System dynamics
• Some Basic Concepts in System Dynamics Jay W. Forrester, MIT Sloan School
• The Beginning of System Dynamics Jay W. Forrester, MIT Sloan School
• The Limits to Growth

We will only do the simple cases

Exercise: eating-duplication system

We want to separate eating from duplication

There are two processes and three items

• food
• cells
• fat cells

Draw this model with pen and paper