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Mathematically Controlled Comparisons

Rui Alves

Ciencies Mediques Basiques

Universitat de Lleida

Outline

Design Principles

Classical Mathematically Controlled Comparisons

Statistical Mathematically Controlled Comparisons

What are design principles?

Qualitative or quantitative rules that explain why certain designs are recurrently observed in similar types of systems as a solution to a given functional problem

Exist at different levels Nuclear Targeting Sequences

Operon

Gene 1 Gene 2 Gene 3

Alternative sensor design in two component systems

S

S*

R*

R

Q1 Q2

Monofunctional Sensor Bifunctional Sensor

S

S*

R*

R

Q1 Q2

Alternative sensor design in two component systems

X3

X1

X2

X4

X5 X6


X3

X1

X2

X4

X5 X6

Why two types of sensor?

Why do two types of sensor exist?

Hypothesis:Random thing

Alternative hypothesis:There are physiological characteristics in the

systemic response that are specific to each type of sensor and that offer selective advantages under different functionality requirements

X3

X1

X2

X4

X5 X6

How do we test the alternative hypothesis?1 – Identify functional criteria that have physiological relevance

i) Appropriate fluxes & concentrations

ii) High signal amplification

iii) Appropriate response to cross-talk

iv) Low parameter sensitivity

v) Fast responses

vi) Large stability margins X5

X2

Time

[X2]

Decrease in X5Fluctuation

in X2

Functionality criteria for effectiveness

Appropriate fluxes & concentrations High signal amplification Appropriate response to cross-talk Low parameter sensitivity Fast responses Large stability margins

How to test the alternative hypothesis?1 – Identify functional criteria that have physiological relevance

2 – Create Mathematical models for the alternativesS-system has analytical steady state solutionAnalytical solutions → General features of the model that

are independent of parameter values

X3

X1

X2

X4

X5 X6

A model with a monofunctional sensor

3/ 1/

4 / 2 /

dX dt dX dt

dX dt dX dt

13 15 11 141 11/ 3 5 1 4g g h hdX dt X X X X

Monofunctional Sensor

21 26 242

2222 / 21 6 4g g g hdX dt X X X X

X3

X1

X2

X4

X5 X6

A model with a bifunctional sensor

3/ 1/

4 / 2 /

dX dt dX dt

dX dt dX dt

13 15 11 121 11/ 3 5 1 2g g h hdX dt X X X X

Bifunctional Sensor

21 26 224 22

2322 / 1 6 4 2 3g h hg g XdX dt X X X X

Approximating the conserved variables

3 4

2 7

3 42 0 7 0

3 7 0 4 0 0

3 1

3 / 1

/ 3 1 * / 3

f f

f f

X X X

X X X

f X X f X X

13 15 11 141 11/ 3 5 1 4g g h hdX dt X X X X

21 26 242

2224 / 21 6 4g g g hdX dt X X X X

Monofunctional Sensor 1 2

1 8

1 21 0 8 0

1 8 0 2 0 0

4 2

4 / 2

/ 4 2 / 4

f f

f f

X X X

X X X

f X X f X X

X3

X1

X2

X4

X5 X6

The S-system equations

13 143 4 15 11 1 21 7 1 1 81/ 1 5 1 2

g hf f g h f fdX dt X X X X X X

2421 26 1 22 1

22282 6 2 2/ 1

gg g f hfdX dt X X X X X

Monofunctional Sensor

Bifunctional Sensor 13 143 4 15 11 1 2

1 7 1 1 81/ 1 5 1 2g hf f g h f fdX dt X X X X X X

2421 26 1 22 1 8

233 4 222 7 1 22 / 1 6 2

gg g f f hf f hdX dt X X X X X X X

S-systems have analytical solutions

22 1 12 2 22 15 12 26

22 17 12 27 7 22 18 12 28 8

11 22 12 21

11 2 21 1 21 15 11 26

21 17 11 27 7 21 18 11 28 8

[ 5] [ 6]

[ ] [ ][ 1]

[ 5] [ 6]

[ ] [ ][ 2]

a b h b a g Log X h g Log X

a g h h Log X a h h g Log XLog X

a a h a

a b a b a g Log X a g Log X

a g a h Log X a h a g Log XLog X

11 22 12 21

11 11 11 21 21 21 22 22 22

a a h a

a g h a g h a g h

04/20/23 16

Analytical solutions are nice!!

Calculating analytical expressions for the gains of the dependent variables with respect to independent variables (Signal amplification) is possible

The same for sensitivity to parameters

The same for other magnitudes

Calculating gains is taking derivatives

11 2 21 1 21 15 11 26

21 17 11 27 7 21 18 11 28 8

11 22 12 21

21 15

11 22 12 21

[ 5] [ 6]

[ ] [ ]

[ 2, 5][ 5]


a g a h Log X a h a g Log Xd

a a h a

L X XdLog X

a g

a a h a

11 2 21 1 21 15 11 26

21 17 11 27 7 21 18 11 28 8

11 22 12 21

11 26

11 22 12 21

[ 5] [ 6]

[ ] [ ]

[ 2, 6][ 6]


a g a h Log X a h a g Log Xd

a a h a

L X XdLog X

a g

a a h a


Appropriate fluxes & concentrations High signal amplification Appropriate response to cross-talk Low parameter sensitivity Large stability margins Fast responses

Outline

Design Principles



How to test the alternative hypothesis?1 – Identify functional criteria that have physiological relevance

2 – Create Mathematical models for the alternatives S-system has analytical steady state solutionAnalytical solutions → General features of the model that

are independent of parameter values

3 – Compare the behavior of the two models with respect to the functional criteria defined in 1

Comparison must be made appropriately, using Mathematically Controlled Comparisons

How to compare the inherent differences between designs?

X3

X1

X2

X4

X5 X6

X3

X1

X2

X4

X5 X6

13 15 11 141 11/ 3 5 1 4g g h hdX dt X X X X

21 26 242

2222 / 21 6 4g g g hdX dt X X X X

13 15 11 141 11/ 3 5 1 4g g h hdX dt X X X X

21 26 224 22

2322 / 1 6 4 2 3g h hg g XdX dt X X X X

Internal Contraints: Corresponding parameters in processes that are identical have the same values in both designs

How to compare the inherent differences between designs?

X3

X1

X2

X4

X5 X6

X3

X1

X2

X4

X5 X6

21 26 2 '2242 22 / 1 6 4 ' 2g g hgdX dt X X X X 21 26 224 2

223

22 / 1 6 4 2 3g h hg g XdX dt X X X X

External constraints:’2 and h’22 are degrees of freedom that the system can use to overcome the loss of bifunctionality.

Reference System

04/20/23 23

How do we implement external contraints?

Identify variables that are important for the physiology of the system

Choose one of those variables Equal it between the reference system and the

alternative system Calculate what the value that leads to such

equivalence is for the primed parameter

Partial controlled comparisons There can be situations where the physiology is

not sufficiently known → Not enough external contraints for all parameters

There can be interest in determining the effect of different sets of physiological contrainst upon parameter values→ Alternative sets of external constraints

Implementing external constraintsChoose Functional Criteria so that the value of the primed parameters can be fixed.

External Constraint 1:

Both systems can achieve the same steady state concentrations AND fluxes

Fixes 2’

Implementing external constraintsChoose Functional Criteria so that the value of the primed parameters can be fixed.

External Constraint 2:

Both systems can achieve the same total signal amplification

Fixes h22’

Studying physiological differences of alternative designs

31 34 32 33 363 3 1 4 3 2 3 6

...

...

g g h h hX X X X X X

04/20/23 27

'34 32 33 363 3 4 3 2 3 6

...

'

...

g h h hX X X X X

AM

Q

AB

Q

AB

AM

Q

1

Comparing concentrations and fluxes

Concentrations and fluxes can be the same in the presence of a bifunctional sensor or of a monofunctional sensor

Comparing signal amplification

Signal amplification is larger in the system with bifunctional sensor

+ - - ++

+ + + - + - ++ Property in Reference system

Property in Alternative system

Comparing cross-talk

Sensitivity to cross talk is higher in the system with monofunctional sensor

+

+ + -

-

+ +-

-

+ Property in Reference system


Comparing sensitivities

Sensitivities can be larger in either system, depending on which sensitivity and on parameter values.

Comparing stability margins

The system with a monofunctional sensor is absolutely stable and has larger stability margins than the system with a bifunctional sensor

Comparing transient times

Undecided

31 34 32 33 363 3 1 4 3 2 3 6

...

...

g g h h hX X X X X X '34 32 33 363 3 4 3 2 3 6

...

'

...

g h h hX X X X X

LinearizeLinearize

Calculate analytical solution

Calculate analytical solution


Undecided


Appropriate Concentrations → Both Systems = Appropriate Fluxes → Both Systems = Signal amplification → Bifunctional larger Cross-talk amplification → Bifunctional smaller Margins of stability → Bifunctional smaller Sensitivities to parameter changes → Undecided Fast transient responses → Undecided

Physiological predictions

Bifunctional design lowers X6 signal amplification prefered when cross-talk is undesirable.

Monofunctional design elevates X6 signal amplification prefered when cross-talk is desirable.

Questions

What happens when ratios depend on parameter values to be larger or smaller than one?

When the ratios are always larger or smaller than one, independent of parameter values, how much larger or smaller are they, on average?

A solution to both problems


Outline

Design Principles



Alternative sensor design in Two Component Systems

X3

X1

X2

X4

X5 X6


X3

X1

X2

X4

X5 X6


Appropriate Concentrations → Both Systems = Appropriate Fluxes → Both Systems = Signal amplification → Bifunctional larger Cross-talk amplification → Bifunctional smaller Margins of stability → Bifunctional smaller Sensitivities to parameter changes → Undecided Fast transient responses → Undecided

Quantifying the differences

To find out how much bigger or smaller or to decide whether an undecided ratio is bigger or smaller than one we have to plug in numbers into the equations

Statistical controlled comparisons

Interested in a specific system from a specific organism: Plug in values and calculate the quantitative

differences Interested in large scale analysis

Large scale sampling of parameter and independent variable space followed by calculation of properties and statistical comparison

Statistical controlled comparisons

Parameters: s, s gs, hs

Independent Variables X5, X6, X7, X8

Basic sampling

Random number generator

L1L’1

Sample in Log space

X5X6Random number generator

[-L’’1,X5,L’’’1], ...

Sample in Log space

gg2Random number generator

[-5,g1,0], [0,g1,5] ...Sample

Importance sampling


Sample1 [-L1,1,L’1]

Normal, Bessel,…Uniform

Filters:

Positive Signal Amplification

Stable Steady State

Fast Response Times

Calculate Values for systemic properties

YesKeep set

NoDiscard set

Warnings about the filters in sampling

Make sure that both the reference and the alternative systems fullfil the filters

Make sure that the sign for the kinetic orders calculated through the external constraints is as it should be

Problems with the sampling

Systems with bifurcations in flux

Systems with conservation relationships

Systems with bifurcations in flux

13 12 11 121 1 3 5 1 1 4/ g g h hdX dt X X X X X3

X1

X2

X4

X5 X6

1 211

1 2

1 2v v

ss

v g v gh

v v

v1 v2

The measure of the set of parameter values within

parameter space that is consistent (generates a steady state that is consistent with v1 and v2) is 0

Systems with moiety conservation

11 123 4 15 11 1 21 1 7 1 5 1 1 1 8 2/

g hf f g h f fdX dt X X X X X X X3

X1

X2

X4

X5 X6

3 4

3 2 7 1

3 42 30 7 10

3 7 30 4 10 30

/

/ /

f f

f f

X X X

X X X

f X X f X X

The measure of the set of parameter values within

parameter space that is consistent (generates a steady state that is consistent with v1 and v2) is 0

Consistent sampling

Sampling Result Space

Sampling without approximating moiety relationships or aggregating fluxes (AMRAF)

Sampling result space

i-2i-n


L1L’1

Sample in Log space

X5X6, X1,X2,X3,X4


[-L’’1,X5,L’’’1], ...

Sample in Log space

gg2Random number generator

[-5,g1,0], [0,g1,5] ...Sample

N rate constants are left to be calculated from the values of the remaining sampled parameters

and variable

N is the number of equations in the ODE system

04/20/23 53

Sampling GMA systems

13 121

114112 11112 11

1/ 7 1 5

1 1 8 2

g g

hh h

dX dt X X X

X X X X

Using GMA form/Don’t

approximate moeity

Sample & Solve Steady State Numericaly

Effects of constraints on parameter values

Using this type of filters allows

Studying which physiological contrainst are important in selecting the range of values for a given parameter

Studying how different contrainst interact with each other to generate a given parameter value distribution

Effect of filters on output parameter distribution

Parameter

High gains

Parameter

Stable SS

Bothf f

Effect of input ditributions on output distributions

Parameter Parameter

Filters

Parameter

Filters

Parameter

f

f f

f

Effects on parameter distributions

Uncontrained SamplingFully Contrained Sampling

Analyzing the results

Set of parametervalues

Set of Steady State properties

Reference

Set of Ratios

Property

Rat

io

1

Using point measures

Property

Rat

io

1

Compare Means, Medians, sd,

quantiles

Alternative System Reference System

Reference system

has higher values

Reference system

has lower values

High Threshold

Using distributions

Property

Rat

io

1

Property, R

f

Property, A

f

Property, A

f

Property, R

f

Low Threshold

Moving median plots

Property

Rat

io1

Property

Rat

io

1

Effect of input ditributions on properties and ratios

Parameter

fCalculation

Parameter

f Calculation

1

Property

Rat

io

1

Property

Rat

io

Sensor logarithmic gains

Y-Axis: Property in Reference system


Regulator logarithmic gains



Sensitivities

04/20/23 66

Stability




Compare

31 34 32 33 363 3 1 4 3 2 3 6

...

...

g g h h hX X X X X X '34 32 33 363 3 4 3 2 3 6

...

'

...

g h h hX X X X X

Numerically Solve ODEs

Numerically Solve ODEs

Response times

Y-Axis: Property in Alternative system

Property in Reference system

Quantifying decided criteria

Average signal amplification → Bifunctional larger (up to 10%)

Average cross-talk amplification → Bifunctional smaller (up to 4%)

Average margins of stability → Bifunctional smaller (up to 4%)

04/20/23 70

Quantifying undecided criteria

Average Sensitivities → Difference smaller than 0.5%

Average transient responses → Bifunctional faster up to 10%

04/20/23 71

Summary

Control ComparisonsAnalyticalStatistical

Two component systemsBifunctional sensor better at buffering against

cross talkMonofunctional sensor absolutely stable and

better integrator of cross talk.

04/20/23 72

Bibliography

Alves & Savageau 2000, 2001, Bioinformatics. Alves & Savageau 2003, Mol Microbiol. Schwacke & Voit 2004 Theor Biol. Med.

Modelling

04/20/23 73

A note on hysteresis

Signal

Res

pons

e

Unstable steady state

At least three steady states must coexist for hysteresis to be a possibility

Hysteresis in classical TCS

The module with a monofunctional sensor has a steady state that is absolutely stable

The module with a bifunctional sensor has unstable steady states→ Hysterisis?

m=1

n=1

1 5 7 1 11 1 8 2 12 1

26 2211 1 8 2 22 6 8 2 21 2 7 1 22 2

8 2 4

7 1 3

0

0

m

ng h

X X X X X X X

X X X X X X X X X X

X X X

X X X

1 5 71

1 5 11 8 2 12

261 5 711 8 2 22 6 8 2

1 5 11 8 2 12

1 5 721 2 7 22 2

1 5 11 8 2 12

8 2 4

7 1 3

0 g

X XX

X X X

X XX X X X X

X X X

X XX X X

X X X

X X X

X X X

21 1 0a X bX c

At most 2 steady states

Hysteresis requires 3 steady states

Therefore, no hysteresis

Finding the steady state

1 5 7 1 11 1 8 2 12 1

2611 1 8 2 22 6 8 2 21 2 7 1 2 22 2

8 2 4

7 1 3

0

0 /

m

g

X X X X X X X

X X X X X X X X X K X X

X X X

X X X


1 5 71

11 8 2 12 1 5

261 5 711 8 2 22 6 8 2

11 8 2 12 1 5

1 5 721 2 7 2 22 2

11 8 2 12 1 5

8 2 4

7 1 3

0

/

g

X XX

X X X

X XX X X X X

X X X

X XX X K X X

X X X

X X X

X X X


3 22 2 2 0a X bX cX d

Three positive non-multiple roots must exist if hysteresis exists

261 5 711 8 2 22 6 8 2

11 8 2 12 1 5

1 5 721 2 7 2 22 2

11 8 2 12 1 5

/ 0

gX XX X X X X

X X X

X XX X K X X

X X X

a, b, c and d are sums and differences of products of positive parameters and independent variables

Analysis of the roots

1 2 3

3 23 2 1 1 2 2 3 1 3 1 2 3( ) ( )

X r X r X r

X X r r r X r r r r r r r r r

If all roots are real and positive, the coefficients have alternating signs

Necessary but not sufficient condition (2 negative roots can have the same pattern, depending on their

values)

_+_+

g268 22 6 1 5 12 11 8 1 5 11 7d = KX X X + X + X X


0 ?

? 0

Sign a Sign c

Sign c Sign d

g2611 22 6 22a = - X + 1Sign a

b = BIG MESS Sign b depends on parameter values

c = BIG MESS Sign c dependson parameter values

No alternating signs No three steady states

No hysteresis

04/20/23 80

No hysteresis in TCS

Thus, neither the monofunctional nor the bifunctional module can, in principle exhibit hysteresis

Summary

Control ComparisonsAnalyticalStatistical

Two component systemsBifunctional sensor better at buffering against

cross talkMonofunctional sensor absolutely stable and

better integrator of cross talk.

Acknowledgments

Mike Savageau Albert Sorribas Armindo Salvador

PGDBM JNICT FCT Spanish Government Portuguese Government NIH (Mike Savageau) DOD (ONR) (Mike Savageau)

Sampling without AMRAF

13 151

11 141

1/ 3 5

1 4

g g

h h

dX dt X X

X X

13 121

112 111 11412 11

1/ 3 5

1 1 4

g g

h h h

dX dt X X

X X X


approximating moiety relationships or aggregating fluxes

S-system form without

approximating Moiety conservation

relationships

Using GMA form/Don’t approximate moeity


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