Class 17: Exponential Models

Methodology of Scientific Research

Andrés Aravena, PhD

April 6, 2023

How cells grow

Cells in a culture grow every day

We want to know the number of cells every day: \(\text{ncell(t)}\)

Here \(t\) is the time in days.

We start with an initial number of cells, that we call \(\text{initial}\)

Each day, the number of cells increases by a factor \(\text{rate}\)

\[\text{ncell(t)} = \text{initial} \cdot \text{rate}^{t}\]

How can we model it?


We cannot see what happens when values are small

Logarithmic scale (“semi–log”)

We can see better using a logarithmic vertical scale

Relation of Body size v/s metabolic rate

Kleiber’s Law

(Physiological Reviews 1947 27:4, 511-541)

Same in semi–log scale

Third plot: log–log


What is a logarithm?

We need very little math: arithmetic, algebra, and logarithms

Just remember that if \(x=p^m\) then \[\log_p(x) = m\] For example \[\log_{10}(10000) = 4\]

We can change the base

If we use another base, for example \(q\), then \[\log_q(x) = \log_p(x) /\log_p(q)\] For example \[\begin{aligned} \log_2(10000) &= \log_{10}(10000)/\log_{10}(2)\\ \log_2(10000) &= 4/\log_{10}(2)\\ 13.28771 &= 4 / 0.30103 \end{aligned}\]

We can choose the best base

So if we use different bases, there is only a scale factor

The “easiest” one is natural logarithm

If \[x=\exp(m)\] then \[\log(x)=m\]

Other things about logarithms

  • They only work with positive numbers. Not with 0

  • If \(x=p\cdot q\) then \[\log(x)=\log(p)+\log(q)\]

  • If \(x=a^m\) then \[\log(x)=m\log(a)\]

Linear models can be used in three cases

Basic linear model \[y=A+B\cdot x\] Exponential change (Initial value and growth Rate) \[y=I\cdot R^x\] Power law (Constant and Exponent) \[y=C\cdot x^E\]

In other words

Basic linear model \[y=A+B\cdot x\] Exponential: if \(y=I\cdot R^x\) then \[\log(y)=\log(I)+\log(R)\cdot x\] Power of \(x\): if \(y=C\cdot x^E\) then \[\log(y)=\log(C)+E\cdot\log(x)\]

Which one to use?

The easiest way to decide is to

  • draw several plots, placing log() in different places,
  • see which one seems more like a straight line

For example, let’s analyze data from Kleiber’s Law

Exercise: Make all plots. Which plot seems more “straight”?

Which plot seems more “straight”?

The plot that seems more straight line is the log–log plot

Therefore we need a log–log model.

\[\log(\text{kcal})=β_0 + β_1 \cdot \log(\text{kg})\]

What is the interpretation of these coefficients?

If \(\log(\text{kcal})=4.21 + 0.756\cdot \log(\text{kg})\) then \[\text{kcal}=\exp(4.21) \cdot \text{kg}^{0.756} =67.1 \cdot \text{kg}^{0.756}\]


  • For a 1kg animal, the average energy consumption is \(\exp(4.21) = 67.1\) kcal
  • The energy consumption increases at a rate of \(0.756\) kcal/kg.

This is Kleiber’s Law

“An animal’s metabolic rate scales
to the ¾ power of the animal’s mass”.

Google it

Prediction: What is wrong here?

animal kg kcal predicted
Mouse 0.021 3.6 1.285
Rat 0.282 28.1 3.249
Guinea pig 0.410 35.1 3.532
Rabbit 2.980 167.0 5.031
Cat 3.000 152.0 5.036
Macaque 4.200 207.0 5.291
Dog 6.600 288.0 5.632
animal kg kcal predicted
Goat 36.0 800 6.915
Chimpanzee 38.0 1090 6.955
Sheep ♂ 46.4 1254 7.106
Sheep ♀ 46.8 1330 7.113
Woman 57.2 1368 7.265
Cow 300.0 4221 8.517
Young cow 482.0 7754 8.876

Undoing the logarithm

We want to predict the metabolic rate, depending on the weight

The independent variable is \(\text{kg}\), the dependent variable is \(\text{kcal}\)

But our model uses only \(\log(\text{kg})\) and \(\log(\text{kcal})\)

So we have to undo the logarithm, using \(\exp()\)

Correct formula for prediction

animal kg kcal predicted
Mouse 0.021 3.6 3.616
Rat 0.282 28.1 25.762
Guinea pig 0.410 35.1 34.185
Rabbit 2.980 167.0 153.113
Cat 3.000 152.0 153.889
Macaque 4.200 207.0 198.458
Dog 6.600 288.0 279.287
animal kg kcal predicted
Goat 36.0 800 1007
Chimpanzee 38.0 1090 1049
Sheep ♂ 46.4 1254 1220
Sheep ♀ 46.8 1330 1228
Woman 57.2 1368 1429
Cow 300.0 4221 5001
Young cow 482.0 7754 7157

Visually (log–log scale)

Visually (linear scale)

In the paper

Exponential growth in Science and Technology

Moore’s Law

A idea from 1965, by George Moore (Intel)

The simple version of this law states that processor speeds will double every two years

More specifically, “the number of transistors on a CPU would double every two years”

(see paper)

Moore’s data and model

Intel says

Real data: Number of transistors v/s year

Semi–log scale: Number of transistors v/s year

Computers get faster and faster

Definition of FLOPS

A ‘flops’ is a floating point operation per second

In simple words, is the number of multiplications per second that a computer can do

Same happens with DNA

Cost of sequencing human genome

Cost of sequencing human genome

Months since Sept 2000

Data for practice

What does this mean for you?

The Robots Are Coming

John Lanchester

An article in the “London Review of Books”

He tells this story

  • 1992 Russo-American moratorium on nuclear testing
  • Just after the Cold War
  • 1996 Computer simulations to design new weapons
  • Needed more computing power than could be delivered by any existing machine

The fastest computer of that time

  • USA designed ASCI Red, the first supercomputer doing over one teraflop

  • In 1997, ASCI Red did 1.8 teraflops

  • It was the most powerful supercomputer in the world until about the end of 2000.

In his article, John Lanchester says

“I was playing on Red only yesterday – I wasn’t really, but I did have a go on a machine that can process 1.8 teraflops.

“This Red equivalent is called the PlayStation3: it was launched by Sony in 2005 and went on sale in 2006.

Comparison of supercomputers

Red characteristics

  • the size of a tennis court
  • used as much electricity as 800 houses
  • cost US$55 million.

The PS3

  • fits under the TV,
  • runs off a normal power socket,
  • costs less than £200.

Things are changing fast

“[In 10 years], a computer able to process 1.8 teraflops went from being something that could only be made by the world’s richest government […], to something a teenager could expect [as a gift].

That was 15 years ago

The Robots Are Coming