# Scientific Models

Table of Contents

#### Model

• A model is a simple representation of complex phenomena.
• Scientific models are used for visualizing, explaining, and forecasting.

#### Examples

##### Double Helix Model of the DNA Molecule
• In 1953 Francis Crick and James Watson proposed a theory of the structure of the DNA molecule in their paper A Structure for Deoxyribose Nucleic Acid
• Their theory was correct.
• They also made a physical model of the molecule.
• DNA can also be represented by
• A structural formula
• A ball-and-stick model, physical or virtual.
• A diagram.
##### Quantum Mechanical Atomic Model
• The model is derived from Quantum Mechanics, using approximate computational methods.
• “The Schrödinger equation cannot be solved precisely for atoms with more than one electron, because of the number of particles and the variety of forces involved.” (Britannica)
• In the Quantum Model electrons occupy orbitals rather than travel in orbits, like planets.
• An orbital is a three-dimensional region within which there is a 95 percent probability of observing an electron.
• Orbitals have different shapes, sizes, and energy levels; and are inhabited by at most two electrons.
##### Numerical Weather Prediction
• A numerical weather prediction uses a mathematical model that makes predictions based on:
• current weather conditions at different locations e.g. temperature, humidity, precipitation, air pressure, wind, cloud cover.
• equations relating the weather variables, e.g. the Ideal Gas Law
• a database of the model’s past performance under similar weather conditions.
• The prediction is made in time steps.
• The current weather is used to predict the weather in 10 minutes, for example. The weather in 10 minutes is used to predict the weather in 20 minutes. And so on.
##### Model Organisms
• Studies of model organisms (e.g. mice, fruit flies, worms) are the basis for drawing conclusions by analogy about higher-order organisms, typically human beings.
• For example, Thomas Hunt Morgan inferred from his experiments breeding fruit flies that human genes were located in a series on chromosomes.
##### Nest Egg Model
• A nest egg model estimates how long your nest egg lasts in retirement.
• Suppose
• your nest egg is \$500,000
• you’re 65
• the year is 2020
• the annual rate of return on your nest egg is 6 percent.
• the annual inflation rate is 2 percent
• you withdraw \$2,500 monthly, increased annually to keep up with inflation.
• The Mathematica program below estimates your nest egg will last until you’re 90 years old.

Mathematica Program for Nest Egg Model

• balance = 500000;
• age = 65;
• currentmonth = 0;
• currentyear = 2020;
• ror[x_] := x * 1.06;
• inflation[x_] := x * 1.02;
• month = {2500, 2500, 2500, 2500, 2500, 2500, 2500, 2500, 2500, 2500, 2500, 2500};
• Print[currentyear, ” “, currentmonth, ” “, balance]
• While[balance > 0,
• currentmonth = currentmonth + 1;
• yearend = If[currentmonth == 13, True, False];
• currentyear = If[yearend, currentyear + 1, currentyear];
• currentmonth = If[yearend, 1, currentmonth];
• age = If[yearend, age + 1, age];
• balance = If[yearend, ror[balance], balance];
• Do[month[[n]] = If[yearend, inflation[month[[n]]], month[[n]]], {n,
• 12}];
• balance = balance – month[[currentmonth]];
• If[currentyear == 2100, Break[]];
• ];
• {currentyear, currentmonth, balance, month[[currentmonth]], age}
• {2045, 10, -2835.25, 4101.51, 90}
##### Supply and Demand Model
• Law of Demand
• As the price of a good or service increases, the quantity demanded decreases, other things being equal.
• Other things:
• population size, consumer incomes, tastes, price of other products
• The Demand Curve has a negative slope
• Law of Supply
• As the price of a good or service increases, the quantity supplied increases, other things being equal.
• Other things:
• size of the industry, technological progress, price of inputs, prices of related outputs
• The Supply Curve has a positive slope.
• Law of Supply and Demand
• The equilibrium price of a good or service, along with the equilibrium quantity supplied or demanded, is where the demand and supply curves intersect, other things being equal.