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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 2Gene 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 Monofunctional Sensor Bifunctional Sensor 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 X 5 Fluctuation 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 alternatives S-system has analytical steady state solution Analytical solutions → General features of the model that are independent of parameter values

X3 X1 X2 X4 X5 X6 A model with a monofunctional sensor Monofunctional Sensor

X3 X1 X2 X4 X5 X6 A model with a bifunctional sensor Bifunctional Sensor

Approximating the conserved variables Monofunctional Sensor X3 X1 X2 X4 X5 X6

The S-system equations Monofunctional Sensor Bifunctional Sensor

S-systems have analytical solutions

4/12/ 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

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

Outline Design Principles Classical Mathematically Controlled Comparisons Statistical Mathematically Controlled Comparisons

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 solution Analytical 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 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 External constraints:  ’ 2 and h’ 22 are degrees of freedom that the system can use to overcome the loss of bifunctionality. Reference System

4/12/ 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 constraints Choose 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 constraints Choose 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 h 22 ’

Studying physiological differences of alternative designs 4/12/ AMAM Q ABAB Q ABAMABAM 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 Property in Alternative 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 Linearize Calculate analytical solution

Comparing transient times Undecided

Functionality criteria for effectiveness 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 Statistical Mathematically Controlled Comparisons

Outline Design Principles Classical Mathematically Controlled Comparisons Statistical Mathematically Controlled Comparisons

Alternative sensor design in Two Component Systems X3 X1 X2 X4 X5 X6 Monofunctional Sensor Bifunctional Sensor X3 X1 X2 X4 X5 X6

Functionality criteria for effectiveness 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    L1  L’1  Sample in Log space X5  X6  Random number generator [-L’’1,X5,L’’’1],... Sample in Log space g  g2  Random number generator [-5,g1,0], [0,g1,5]... Sample

Importance sampling Random number generator Sample 11 [-L1,  1,L’1] Normal, Bessel,… Uniform Filters: Positive Signal Amplification Stable Steady State Fast Response Times Calculate Values for systemic properties Yes Keep set No Discard 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 X3 X1 X2 X4 X5 X6 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 X3 X1 X2 X4 X5 X6 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-2  i-n Random number generator    L1  L’1  Sample in Log space X5  X6, X1, X2,X3,X4 Random number generator [-L’’1,X5,L’’’1],... Sample in Log space g  g2  Random 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

4/12/ Sampling GMA systems 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 Both f f

Effect of input ditributions on output distributions Parameter Filters Parameter Filters Parameter f f f f

Effects on parameter distributions Uncontrained Sampling Fully Contrained Sampling

Analyzing the results Set of parameter values Set of Steady State properties Reference Set of Ratios Property Ratio 1

Using point measures Property Ratio 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 Ratio 1 Property, R f Property, A f f Property, R f Low Threshold

Moving median plots Property Ratio 1 Property Ratio 1

Effect of input ditributions on properties and ratios Parameter f Calculation Parameter f Calculation 1 Property Ratio 1 Property Ratio

Sensor logarithmic gains Y-Axis: Property in Reference system Property in Alternative system

Regulator logarithmic gains Y-Axis: Property in Reference system Property in Alternative system

Sensitivities

4/12/ Stability Y-Axis: Property in Reference system Property in Alternative system

Comparing transient times Compare 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%)

4/12/ Quantifying undecided criteria Average Sensitivities → Difference smaller than 0.5% Average transient responses → Bifunctional faster up to 10%

4/12/ Summary Control Comparisons  Analytical  Statistical Two component systems  Bifunctional sensor better at buffering against cross talk  Monofunctional sensor absolutely stable and better integrator of cross talk.

4/12/ Bibliography Alves & Savageau 2000, 2001, Bioinformatics. Alves & Savageau 2003, Mol Microbiol. Schwacke & Voit 2004 Theor Biol. Med. Modelling

4/12/ A note on hysteresis Signal Response 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 At most 2 steady states Hysteresis requires 3 steady states Therefore, no hysteresis Finding the steady state

Three positive non-multiple roots must exist if hysteresis exists a, b, c and d are sums and differences of products of positive parameters and independent variables

Analysis of the roots 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) _ + _ +

Finding the steady state No alternating signs No three steady states No hysteresis

4/12/ No hysteresis in TCS Thus, neither the monofunctional nor the bifunctional module can, in principle exhibit hysteresis

Summary Control Comparisons  Analytical  Statistical Two component systems  Bifunctional sensor better at buffering against cross talk  Monofunctional 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 Sample & Solve Steady State Numericaly approximating moiety relationships or aggregating fluxes S-system form without approximating Moiety conservation relationships Using GMA form/Don’t approximate moeity Sample & Solve Steady State Numericaly