Chapter 4Equations

(Formal Models)

𝑁𝑡=𝑁0𝑒¿¿ +

𝑊 𝑠

𝑉 𝑖=𝑃𝑠 ∙

𝑆𝑠𝑉 𝑖❑

2 /3 ∙𝑇 𝑠

𝑉 𝑖❑1/3𝐻=2.094𝑀𝐸𝐼 0.461

𝑡=1.3+𝐿𝐻2𝑂𝑚=𝐿 ∙ 𝑡−𝐷𝑒− 𝐼 ∙𝑡

Femur lengt=𝑏(body length)α1𝑁𝑑𝑁𝑑𝑡 =𝑟 − 𝛽𝑃

𝐼=𝐼𝑚𝑎𝑥 (1−𝑒− 𝜁( p− p′ ))

�̇�=𝛼𝑀 𝛽

Expressing Biological Concepts

Data

Verbal

Graphical Formal𝑁𝑡=𝑁0𝑒0.6 ∙𝑡

“Bacterial growth rate is exponential”Most common

Also common

Less common,but necessary

To make calculations,we need an equation

Equations – What’s the good of um?

• They’re the basis for most of quantitative biology

• Have several uses:– Demonstrate how a quantity was calculated

– Make theoretical conclusions

– Test hypotheses

– Scale up experimental scope

How to “read” equations?

• Dissect into components, then reconstruct meaning

• Begin by dissecting equations into terms

• Net energy balance, , is the sum of ingestion, , metabolic losses, , and energy devoted to growth, .

�̇�𝑛𝑒𝑡= �̇�𝑖𝑛𝑔𝑒𝑠𝑡𝑖𝑜𝑛+�̇�𝑟𝑒𝑠𝑝+�̇� h𝑔𝑟𝑜𝑤𝑡, , ,

How to “read” equations?

• Two types of terms:1. Variable quantities [124.22, 135.59, …] Kg• Many possible values

2. Parametric quantities [23.35] Kcal·Kg-1

• Fixed – applied “across measurements”

• e.g. Metabolic rate depends on body size:

�̇�=𝛼𝑀 𝛽

Varia

ble

Varia

ble

Para

met

ric

Para

met

ric�̇�=𝛼𝑀 𝛽

How to “read” equations?

• The identification of quantities identify the meaning of each term

1𝑁𝑑𝑁𝑑𝑡 =�̇�− �̇�, ,

Quantity name Symbol Unit DimensionPopulation size N Bacteria #Time T Hours TRate pop. change dN/dt Bacteria/hr #/TPercapita pop. change N-1dN/dt %/hr T-1

Birth rate B %/hr T-1

Death rate D %/hr T-1

How to “read” equations?

• The identification of quantities identify the meaning of each term

1𝑁𝑑𝑁𝑑𝑡 =�̇�− �̇�, ,

Quantity name Symbol Unit DimensionPopulation size N Bacteria #Time T Hours TRate pop. change dN/dt Bacteria/hr #/TPercapita pop. change N-1dN/dt %/hr T-1

Birth rate B %/hr T-1

Death rate D %/hr T-1

How to “read” equations?

• The identification of quantities identify the meaning of each term

1𝑁𝑑𝑁𝑑𝑡 =�̇�− �̇�, ,

Quantity name Symbol Unit DimensionPopulation size N Bacteria #Time T Hours TRate pop. change dN/dt Bacteria/hr #/TPercapita pop. change N-1dN/dt %/hr T-1

Birth rate B %/hr T-1

Death rate D %/hr T-1

How to “read” equations?

• The identification of quantities identify the meaning of each term

1𝑁𝑑𝑁𝑑𝑡 =�̇�− �̇�, ,

Quantity name Symbol Unit DimensionPopulation size N Bacteria #Time T Hours TRate pop. change dN/dt Bacteria/hr #/TPercapita pop. change N-1dN/dt %/hr T-1

Birth rate B %/hr T-1

Death rate D %/hr T-1

How to “read” equations?

• The identification of quantities identify the meaning of each term

1𝑁𝑑𝑁𝑑𝑡 =�̇�− �̇�, ,

Quantity name Symbol Unit DimensionPopulation size N Bacteria #Time T Hours TRate pop. change dN/dt Bacteria/hr #/TPercapita pop. change N-1dN/dt %/hr T-1

Birth rate B %/hr T-1

Death rate D %/hr T-1

Translation aids comprehension

• Translate to units– Helps visualize meaning

• Translate to dimensions– Helps visualize relation of quantities

• Translate into computations• Translate to graphics

𝑁𝑡=𝑁0𝑒¿¿

𝑁𝑡=𝑁0𝑒¿¿

flies=flies( year−1 ) year

¿=¿❑(T −1 )TUnits:

Dimensions:

Computations:

Graphical:

𝑁𝑡=𝑁0𝑒¿¿

“Exponential growth and

decay of populations”

Formal Verbal

• Use words to connect the formal model to experience

• e.g. Efficiency of carbonate utilization by marine gastropods:

𝑊 𝑠

𝑉 𝑖=𝑃𝑠 ∙

𝑆𝑠

𝑉 𝑖❑

23

∙𝑇 𝑠

𝑉 𝑖❑

13

(c alcificationindex )=( shelldensity )( formindex)(thicknessindex )

𝑊 𝑠

𝑉 𝑖=𝑃𝑠 ∙

𝑆𝑠

𝑉 𝑖❑

23

∙𝑇 𝑠

𝑉 𝑖❑

13

𝑊 𝑠

𝑉 𝑖=𝑃𝑠 ∙

𝑆𝑠

𝑉 𝑖❑

23

∙𝑇 𝑠

𝑉 𝑖❑

13

𝑊 𝑠

𝑉 𝑖=𝑃𝑠 ∙

𝑆𝑠

𝑉 𝑖❑

23

∙𝑇 𝑠

𝑉 𝑖❑

13

1 Parametric: Shell density 2.71 g/cc3 Variable: Shell area mm2

Shell thickness mmShell internal volume mm3

==

Homogeneity of Units

• Equations in biology have units– Terms (sep. =,+,-) have to have the same units– Both sides of an equation must have the same

units• Check: 𝑁𝑡=𝑁0 ∙𝑒𝑟 ∙𝑡

a nts=ants ∙𝑒? ∙ day

a nts=ants ∙𝑒day−1 ∙ day

Homogeneity of Units

• Check:

• Solve for α:

𝐻=2.094𝑀𝐸𝐼 0.461

( fishcatch)(morpho−¿edaphicindex )

lb ∙ acre−1 ∙ year−1 ppm ∙ ft−1≠

𝐻=2.094𝑀𝐸𝐼 0.461

𝐻=𝛼𝑀𝐸𝐼 0.461

Homogeneity of Units

• Solve for α:

• Re-check:

𝐻𝑀𝐸𝐼0.461

=𝛼𝑀𝐸𝐼 0.461

𝑀𝐸𝐼 0.461

𝛼=𝐻 ∙𝑀𝐸𝐼− 0.461

α=2.094 lb ∙ acre−1 ∙ year−1 ppm− 0.461 ∙ ft0.461

𝐻=𝛼𝑀𝐸𝐼 0.461lb ∙ acre−1 ∙ year−1=lb ∙ acre− 1 ∙ year− 1 ppm−0.461 ∙ ft0.461 ( ppm ∙ ft−1 )0.461

lb ∙ acre−1 ∙ year−1= lb ∙ acre− 1 ∙ year− 1 ppm−0.461 ∙ ft0.461 ∙ ppm0.461 ∙ ft− 0.461❑

Homogeneity of Dimensions

𝐻=𝛼𝑀𝐸𝐼 0.461lb ∙ acre−1 ∙ year−1= lb ∙ acre− 1 ∙ year− 1 ppm−0.461 ∙ ft0.461 ( ppm ∙ ft−1 )0.461

lb ∙ acre−1 ∙ year−1= lb ∙ acre− 1 ∙ year− 1 ppm−0.461 ∙ ft0.461 ∙ ppm0.461 ∙ ft− 0.461❑

𝑀𝐿−2𝑇 −1=𝑀 𝐿− 2𝑇 −1% L0.461% L− 0.461❑

• In addition to checking units, it is useful to apply the concept of similarity

• Recognizing dimensional homogeneity is a valuable skill – key for the “reading” of equations

Homogeneity of Dimensions

• Equation for memory retention:

where• m is the memory trace's strength (measured by

recognition)• L is initial strength at the end of learning• t is the retention interval• D is the time decay rate• I is the measure of degree of interference

• Is the equation legit?

𝑚=𝐿 ∙ 𝑡−𝐷𝑒− 𝐼 ∙𝑡