• Computational Thinking
  • Decomposition
  • Pattern Recognition
  • Abstraction
  • Algorithm Design
• Not memory
• Problem solving

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Biology is the study of life

Life is a complex phenomenon

To understand life, biologists invented Systems Theory

Systems Theory was invented in 1948 by the biologist Ludwig von Bertalanffy

Today it is used by engineers, psychologists, managers, and other scientists

It is going to be one of the important ways of thinking in this century

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

• Which are the items?
• Which are the processes?

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.

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

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

Each item 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”

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”

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

The “boxes” of the system represent processes

They do not change with time

The have one constant value called rate

For each process we have a value process_rate

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

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

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.

A chemical reaction combines hydrogen and oxygen to produce water

To make it simple we will assume that the reaction is this: \[2H+O\leftrightarrow H_2O\]

Our system has 3 items:

• Hydrogen
• Oxygen
• Water

We will measure the number of atoms and molecules in moles

The initial quantities are: 1 mole oxygen, 1 mole hydrogen and 0 mole water

We represent hydrogen by the vector H, oxygen by O, and water by W.

Each vector has the number of atoms or molecules for different times

• The speed of reaction1 depends on the number of oxygen atoms and the square of the number of hydrogen atoms \[r_1 = k_1 H^2O\]
• The speed of reaction2 depends on the number of water molecules \[r_2=k_2 W\]
• The constants \(k_1\) and \(k_2\) depend on temperature and other conditions

• The number water molecules increase with the first reaction and decreases with the second reaction \[d\_W = r1 - r2\]
• The number of oxygen atoms reduces with the first reaction and increases with the second \[d\_O = -r1 + r2\]
• The number of hydrogen atoms reduces two times with the first reaction and increases two times with the second \[d\_H = 2*(-r1 + r2)\]

Then the main part of the simulation is

for(i in 2:N) {
    d_W[i] <-    k1*H[i-1]*H[i-1]*O[i-1] - k2*W[i-1]
    d_O[i] <-   -k1*H[i-1]*H[i-1]*O[i-1] + k2*W[i-1]
    d_H[i] <- -2*k1*H[i-1]*H[i-1]*O[i-1] + 2*k2*W[i-1]

    W[i] <- W[i-1] + d_W[i]
    O[i] <- O[i-1] + d_O[i]
    H[i] <- H[i-1] + d_H[i]
}

Create a water_formation function.

Inputs are:

• N: Number of steps in the simulation
• H_ini: initial amount of hydrogen, default 1
• O_ini: initial amount of oxygen, default 1
• W_ini: initial amount of water, default 0
• k1: reaction 1 rate, default 0.1
• k2: reaction 2 rate, default 0.1

One atom of oxygen has 16 times the mass of one atom of hydrogen

Therefore one molecule of water has mass 18

Show that the total mass never changes

Ludwig von Bertalanffy, “General System Theory: Foundations, Development, Applications” New York (1968)

Koch, Ina. “Petri Nets - A Mathematical Formalism to Analyze Chemical Reaction Networks.” Molecular Informatics 29, no. 12 (December 17, 2010): 838–43. doi:10.1002/minf.201000086.

Baez, John C., and Jacob Biamonte. “Quantum Techniques for Stochastic Mechanics,” September 17, 2012. http://arxiv.org/abs/1209.3632.