Sensitivity Analysis

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sensitivity-analysis local-sensitivity global-sensitivity Sobol-indices metabolic-control-analysis

Core Idea

Sensitivity analysis determines how changes in model parameters or inputs affect model outputs, identifying which parameters most strongly influence the system's behavior. Local sensitivity analysis computes partial derivatives of outputs with respect to individual parameters near a specific operating point. Global sensitivity analysis (Sobol indices, Morris screening) explores the full parameter space, accounting for interactions between parameters and nonlinear effects. In metabolic systems, metabolic control analysis (MCA) formalizes sensitivity as flux control coefficients and elasticity coefficients. Sensitivity analysis guides experimental design (measure the parameters that matter most), identifies drug targets (the most sensitive nodes in disease networks), and reveals which model predictions are robust versus parameter-dependent.

Explainer

Every systems biology model, whether an ODE model of signaling dynamics or an FBA model of metabolism, depends on parameters whose values are uncertain. Sensitivity analysis asks: which of these uncertain parameters actually matter for the model's predictions? If the model output barely changes when a parameter varies over its plausible range, that parameter can be fixed at a nominal value without loss. If the output changes dramatically, that parameter is a priority for experimental measurement — and a potential point of biological control or therapeutic intervention.

Local sensitivity analysis is the simplest approach: compute the partial derivative of the output with respect to each parameter at the current operating point. For an ODE model, this can be done analytically (solving the sensitivity equations alongside the state equations) or numerically (perturbing each parameter by a small amount and observing the output change). The result is a sensitivity coefficient for each parameter that quantifies its local influence. In metabolic systems, this concept is formalized as metabolic control analysis (MCA), where the flux control coefficient C_i^J measures how much a fractional change in enzyme i's activity changes the flux J. The summation theorem (all flux control coefficients sum to 1) reveals that metabolic control is shared among enzymes, demolishing the concept of a single rate-limiting step.

Global sensitivity analysis goes further by exploring the entire plausible parameter range. Sobol indices decompose the total variance of the model output into contributions from individual parameters (first-order indices) and from interactions between parameters (higher-order indices). A parameter with a high first-order Sobol index drives substantial output uncertainty on its own; a parameter with a high total-order index (including interactions) may not matter individually but strongly modulates the effect of other parameters. Morris screening provides a cheaper alternative: it samples the parameter space with a design that estimates, for each parameter, the mean and variance of its elementary effect — classifying parameters as negligible (small mean, small variance), linearly important (large mean, small variance), or nonlinearly important or interacting (large variance).

The practical payoff of sensitivity analysis is threefold. For experimental design, it prioritizes which parameters to measure: invest limited experimental resources in the parameters that most influence model predictions. For drug target identification, the most sensitive nodes in a disease network model are the most promising therapeutic targets — perturbations at these nodes produce the largest phenotypic effects. For model robustness assessment, outputs that are insensitive to most parameters are reliable predictions even with uncertain parameter values, while outputs that are highly sensitive to poorly constrained parameters should be interpreted cautiously. Sensitivity analysis transforms a model from a black box into a transparent tool that communicates not just what it predicts, but how confident those predictions are and what drives the uncertainty.

Practice Questions 3 questions

Prerequisite Chain

Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsCombining Like TermsOne-Step EquationsTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIterated Integrals and Fubini's TheoremDouble Integrals in Cartesian CoordinatesDouble Integrals over Rectangular RegionsDouble Integrals in Polar CoordinatesDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsTriple Integrals in Cartesian CoordinatesTriple Integrals in Cylindrical and Spherical CoordinatesChange of Variables and the Jacobian DeterminantApplications of Triple Integrals: Volume and MassVector Fields and Their RepresentationsLine Integrals of Vector FieldsGreen's TheoremSurface Integrals and Flux of Vector FieldsSurface Integrals and Flux of Vector FieldsDivergence Theorem: Flux and OutflowDivergence TheoremElectric FluxGauss's LawConductors in Electrostatic EquilibriumCapacitance and CapacitorsDielectricsDielectric Constant and Relative PermittivityElectric Field Inside Dielectric MaterialsDielectric Materials and PolarizationDielectric Susceptibility and PermittivityEnergy Density in Electric FieldsElectric Current and Current DensityElectrical Resistance and ResistivityOhm's Law and Circuit ElementsElectromotive Force (EMF) and BatteriesKirchhoff's Circuit Laws: Voltage and CurrentDC Circuit Network Analysis MethodsTransient Response in RC CircuitsRC CircuitsLC and RLC CircuitsAC Circuits: FundamentalsImpedance and ReactanceAC Power and ResonanceElectromagnetic WavesThe Electromagnetic SpectrumBlackbody Radiation and Planck's LawPhotoelectric EffectThe Photon: Light as QuantaCompton ScatteringWave-Particle Dualityde Broglie WavelengthHeisenberg Uncertainty PrincipleWavefunction and the Born RuleThe Schrödinger EquationState Vectors and WavefunctionsQuantum SuperpositionQuantum EntanglementBell Theorem and Bell InequalitiesPostulates of Quantum MechanicsScattering TheoryIntroduction to Scattering TheoryPartial Wave Analysis in ScatteringSpin Angular MomentumElectron Spin and Intrinsic Magnetic MomentStern-Gerlach Experiment: Spin Quantization and MeasurementElectron Diffraction and Matter Wave PropertiesDavisson-Germer Experiment: Crystal Diffraction of ElectronsElectron Diffraction and Matter Wave InterferenceWavefunctions and Probability Density InterpretationQuantum Superposition and Linear Combinations of StatesQuantum Operators and ObservablesCanonical Commutation Relations and UncertaintyHeisenberg Uncertainty Principle and Measurement LimitsTime-Independent Schrödinger Equation and EigenvaluesHydrogen Atom in Quantum MechanicsSpectral Lines and Energy TransitionsSelection Rules for Atomic TransitionsLS and jj Coupling Schemes in Multi-Electron AtomsPauli Exclusion Principle and Antisymmetric WavefunctionsElectron Configuration and the Aufbau PrincipleThe Periodic Table and Atomic Electronic StructureThe Periodic TableElectron ConfigurationPeriodic TrendsIonization EnergyIonic BondingLewis StructuresResonance Structures and Delocalized ElectronsResonance and Formal ChargeMolecular Polarity and Dipole MomentsIntermolecular ForcesStates of Matter and Phase Changes: Melting, Boiling, and SublimationGas Laws and the Ideal Gas EquationGas Stoichiometry and Volume-Volume CalculationsThermochemistry and EnthalpyHeat Capacity and CalorimetryEntropy and Molecular DisorderSpontaneity and ΔGEntropy and Gibbs Free EnergyChemical EquilibriumAcid-Base ChemistryOrganic Reaction Mechanisms and Arrow PushingElectrophilic Addition to AlkenesAromaticity and BenzeneDNA StructureCentral Dogma of Molecular BiologyThe Genetic CodeDNA MutationsDNA Repair MechanismsCell Cycle Checkpoints and Cancer PreventionMitotic Spindle Checkpoint and Chromosome SegregationKinetochore Structure and FunctionMitochondria: Structure and FunctionCellular Respiration OverviewGlycolysisPyruvate OxidationThe Krebs Cycle (Citric Acid Cycle)Electron Transport ChainATP Synthesis and Oxidative PhosphorylationPhotosynthesis OverviewTrophic Levels and Food WebsEnergy Flow and Ecological EfficiencyBiogeochemical Cycles: Carbon, Nitrogen, and PhosphorusNutrient Cycling: Phosphorus and Sulfur CyclesPhosphorus Cycling and Freshwater-Marine DifferencesNucleotide Structure and NomenclaturePyrimidine BiosynthesisNucleotide Salvage PathwaysNucleotide Synthesis Pathways (De Novo and Salvage)Transcription Initiation and Gene RegulationPromoters, Enhancers, Silencers, and Cis-Acting ElementsTranscription Factors: DNA Binding and Gene RegulationGene Regulatory NetworksBiological Network AnalysisSignal Transduction NetworksODE Models in BiologyParameter Estimation in Biological ModelsSensitivity Analysis

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