Systems Biology Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Systems Biology Mathematics Ontology within the Kireth Research Semantic Library. It formalises interacting genes, proteins, metabolites, reactions and signalling pathways as mathematical dynamical systems.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Systems Biology Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Biological Mathematics / Systems Biology Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/biological-mathematics/systems-biology-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SystemsBiologyMathematics
 ENTITY_CLASS = MathematicalBiologicalNetworkDynamicsFramework
 PRIMARY_CONTEXT = MathematicalBiology
 DEFINITION = "Systems biology mathematics is the mathematical study of interacting biological components and networks as integrated dynamical systems."
 SYMBOLIC_FORMS = ["dx/dt=S*v(x,p)","dx_i/dt=f_i(x,p)","J_ij=partial f_i/partial x_j"]
}

SCOPE {
 INCLUDE = [
  SystemsBiologyMathematics, SystemsBiology, BiologicalSystem, BiologicalNetwork,
  StateVariable, BiologicalParameter, BiochemicalReactionNetwork, Reaction,
  ReactionRate, Stoichiometry, StoichiometricMatrix, MassActionKinetics,
  EnzymeKinetics, MichaelisMentenKinetics, HillEquation, GeneRegulatoryNetwork,
  GeneExpression, Activation, Repression, ProteinInteractionNetwork,
  SignallingNetwork, SignalTransduction, MetabolicNetwork, MetabolicFlux,
  FluxBalanceAnalysis, SteadyState, Homeostasis, PositiveFeedback,
  NegativeFeedback, FeedForwardLoop, NetworkMotif, Oscillation,
  BiologicalSwitch, Bistability, Multistability, Stability, JacobianMatrix,
  Bifurcation, SensitivityAnalysis, ParameterEstimation, ParameterIdentifiability,
  StructuralIdentifiability, PracticalIdentifiability, ModelReduction,
  TimescaleSeparation, QuasiSteadyStateApproximation, StochasticSystemsBiology,
  ChemicalMasterEquation, GillespieAlgorithm, IntrinsicNoise, ExtrinsicNoise,
  SingleCellVariability, ComputationalSystemsBiology, ModelValidation
 ]
 EXCLUDE_AS_PRIMARY = [
  DescriptiveSystemsBiologyWithoutMathematicalModel,
  StaticBioinformaticsWithoutDynamicSystemSemantics,
  GeneralNetworkTheoryWithoutBiologicalInterpretation
 ]
 DISAMBIGUATION_RULE = "When interacting biological components are represented as an integrated mathematical dynamical system, interpret the context as systems biology mathematics."
}

CORE_CONCEPTS {
 SystemsBiologyMathematics = "Mathematical modelling of interacting biological components as integrated systems."
 BiochemicalReactionNetwork = "Network of biochemical species connected by reactions and kinetic laws."
 StoichiometricMatrix = "Matrix encoding net species changes associated with reactions."
 GeneRegulatoryNetwork = "Network representing regulatory interactions controlling gene expression."
 SignallingNetwork = "Network representing molecular information transfer."
 MetabolicNetwork = "Network of metabolites and biochemical reactions."
 Feedback = "System structure in which outputs influence upstream dynamics."
 ParameterIdentifiability = "Ability to determine model parameters from model structure and observations."
}

RELATIONS {
 SystemsBiologyMathematics --is_child_of--> MathematicalBiology
 SystemsBiologyMathematics --models--> BiologicalSystem
 BiochemicalReactionNetwork --contains--> Reaction
 Reaction --has--> ReactionRate
 StoichiometricMatrix --represents--> ReactionStoichiometry
 MassActionKinetics --defines--> ReactionRate
 MichaelisMentenKinetics --models--> EnzymeKinetics
 HillEquation --models--> CooperativeRegulation
 GeneRegulatoryNetwork --contains--> RegulatoryInteraction
 SignallingNetwork --models--> SignalTransduction
 MetabolicNetwork --contains--> MetabolicReaction
 FluxBalanceAnalysis --uses--> StoichiometricMatrix
 NegativeFeedback --supports--> Homeostasis
 PositiveFeedback --can_support--> Bistability
 JacobianMatrix --supports--> StabilityAnalysis
 Bifurcation --changes--> QualitativeSystemBehaviour
 ParameterIdentifiability --constrains--> ParameterEstimation
 GillespieAlgorithm --simulates--> ChemicalMasterEquation
}

OPERATORS {
 REACTION_NETWORK_ODE(x,S,v) := dx/dt=S*v(x,p)
 MASS_ACTION_RATE(k,x,nu) := v=k*product_i x_i^nu_i
 MICHAELIS_MENTEN(S,Vmax,Km) := v=Vmax*S/(Km+S)
 HILL_ACTIVATION(x,K,n) := h=x^n/(K^n+x^n)
 HILL_REPRESSION(x,K,n) := h=K^n/(K^n+x^n)
 STEADY_STATE(S,v) := S*v(x_star,p)=0
 JACOBIAN(f,x) := J_ij=partial f_i/partial x_j
 LOCAL_STABILITY(J) := stable_if_all_eigenvalues_have_negative_real_part
 FLUX_BALANCE(S,v) := S*v=0
 SENSITIVITY(x,p_j) := s_j=partial x/partial p_j
}

AXIOMS {
 A1 = "Systems biology models require explicit biological entities and interaction meanings."
 A2 = "Reaction-network dynamics require stoichiometry and kinetic assumptions."
 A3 = "Network topology and dynamic parameters jointly determine behaviour."
 A4 = "Steady-state structure does not by itself determine transient dynamics."
 A5 = "Feedback sign and delay can materially alter stability and oscillation."
 A6 = "Deterministic and stochastic molecular models must not be conflated."
 A7 = "Parameter sensitivity and identifiability are distinct properties."
 A8 = "Systems-level conclusions require mechanistic and numerical validation."
}

REASONING {
 R1_CONTEXT = identify(system_components,interactions,state_variables,parameters,data,scale)
 R2_NETWORK = construct(reaction_network,gene_regulatory_network,signalling_network,metabolic_network)
 R3_DYNAMICS = formulate(ODE,SDE,master_equation,constraint_based_model)
 R4_STEADY_STATE = solve(equilibria,flux_balances,homeostatic_states)
 R5_STABILITY = compute(Jacobian,eigenvalues,local_stability)
 R6_FEEDBACK = analyse(feedback_loops,oscillation,bistability,network_motifs)
 R7_PARAMETER = estimate(parameters,identifiability,confidence)
 R8_SENSITIVITY = analyse(local_global_parameter_sensitivity)
 R9_STOCHASTIC = simulate(noise,chemical_master_equation,Gillespie,single_cell_variability)
 R10_VALIDATION = compare(predictions,data,biological_constraints,numerical_convergence)
}

CONSTRAINTS {
 C1 = "Mass-action kinetics require compatible reaction assumptions."
 C2 = "Michaelis-Menten reductions require suitable timescale assumptions."
 C3 = "Hill coefficients are phenomenological unless mechanistically derived."
 C4 = "Flux balance analysis assumes a steady-state mass-balance framework."
 C5 = "Local Jacobian stability does not guarantee global stability."
 C6 = "Large biological networks can be structurally unidentifiable."
 C7 = "Deterministic concentration models may fail at low molecule counts."
 C8 = "Network topology alone does not determine quantitative dynamics."
 C9 = "Model reduction must preserve behaviour relevant to the intended question."
 C10 = "A close fit to data does not uniquely validate a network mechanism."
}

SEMANTIC_EQUIVALENCE {
 "SystemsBiologyMathematics" <=> "MathematicalSystemsBiology"
 "BiochemicalReactionNetwork" <=> "ReactionNetwork"
 "GeneRegulatoryNetwork" <=> "GRN"
 "FluxBalanceAnalysis" <=> "FBA"
 "ChemicalMasterEquation" <=> "CME"
}

VALIDATION {
 V1 = "Biological entities and interactions are explicit."
 V2 = "State-variable units and scales are documented."
 V3 = "Stoichiometric coefficients are internally consistent."
 V4 = "Kinetic laws and parameter units are explicit."
 V5 = "Initial conditions are documented."
 V6 = "Steady states and stability are checked where relevant."
 V7 = "Parameter identifiability is assessed for calibrated models."
 V8 = "Sensitivity and uncertainty are evaluated."
 V9 = "Stochastic simulations report sampling uncertainty."
 V10 = "Model outputs are compared with biological data or constraints."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYSTEMS_BIOLOGY_MATHEMATICS_DEFINITION = 0.99
 REACTION_NETWORK_REASONING = 0.99
 GENE_REGULATION_REASONING = 0.98
 SIGNALLING_REASONING = 0.98
 METABOLIC_REASONING = 0.98
 STABILITY_REASONING = 0.99
 IDENTIFIABILITY_REASONING = 0.98
 STOCHASTIC_REASONING = 0.98
 POLICY = "Reduce confidence when network structure, kinetic laws, parameter units, timescale assumptions, identifiability, molecular counts, initial conditions or validation data are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, SystemsBiologyMathematicsReasoning,
  BiologicalNetworkReasoning, ReactionNetworkReasoning, StoichiometryReasoning,
  MassActionReasoning, EnzymeKineticsReasoning, GeneRegulatoryNetworkReasoning,
  SignallingNetworkReasoning, MetabolicNetworkReasoning, FluxBalanceReasoning,
  FeedbackReasoning, HomeostasisReasoning, StabilityReasoning,
  BifurcationReasoning, ParameterEstimationReasoning, IdentifiabilityReasoning,
  SensitivityReasoning, StochasticSystemsBiologyReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "systems biology mathematics" -> SystemsBiologyMathematics
  "mathematical systems biology" -> SystemsBiologyMathematics
  "reaction network" -> BiochemicalReactionNetwork
  "gene regulatory network" -> GeneRegulatoryNetwork
  "GRN" -> GeneRegulatoryNetwork
  "flux balance analysis" -> FluxBalanceAnalysis
  "FBA" -> FluxBalanceAnalysis
  "chemical master equation" -> ChemicalMasterEquation
 }
}

EXECUTION_RULES {
 E1 = "If interacting biological components are represented mathematically, activate SystemsBiologyMathematicsOntology."
 E2 = "If biochemical reactions determine dynamics, construct a stoichiometric reaction-network model."
 E3 = "If gene activation or repression is central, route to Gene Regulatory Network reasoning."
 E4 = "If signalling interactions are central, route to Signalling Network reasoning."
 E5 = "If steady metabolic fluxes are central, consider Flux Balance Analysis."
 E6 = "If feedback is present, analyse stability, oscillation and multistability."
 E7 = "If molecule counts are low, consider stochastic reaction modelling."
 E8 = "If parameters are inferred from data, require identifiability analysis."
 E9 = "If simplifying a large system, validate model reduction against target behaviour."
}

RECURSION {
 ENTRY = SystemsBiologyMathematics
 EXPAND(SystemsBiologyMathematics) -> [
  SystemsBiology, BiologicalSystem, BiologicalNetwork, StateVariable,
  BiologicalParameter, BiochemicalReactionNetwork, Reaction, ReactionRate,
  Stoichiometry, StoichiometricMatrix, MassActionKinetics, EnzymeKinetics,
  MichaelisMentenKinetics, HillEquation, GeneRegulatoryNetwork, GeneExpression,
  Activation, Repression, ProteinInteractionNetwork, SignallingNetwork,
  SignalTransduction, MetabolicNetwork, MetabolicFlux, FluxBalanceAnalysis,
  SteadyState, Homeostasis, PositiveFeedback, NegativeFeedback, FeedForwardLoop,
  NetworkMotif, Oscillation, BiologicalSwitch, Bistability, Multistability,
  Stability, JacobianMatrix, Bifurcation, SensitivityAnalysis,
  ParameterEstimation, ParameterIdentifiability, StructuralIdentifiability,
  PracticalIdentifiability, ModelReduction, TimescaleSeparation,
  QuasiSteadyStateApproximation, StochasticSystemsBiology,
  ChemicalMasterEquation, GillespieAlgorithm, IntrinsicNoise, ExtrinsicNoise,
  SingleCellVariability, ComputationalSystemsBiology, ModelValidation
 ]
 STOP_CONDITION = "Stop when biological components, network structure, kinetics, dynamics, feedback, parameters, stochasticity and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Biology"
 RELATED_ONTOLOGIES = [
  "Mathematical Biology Ontology",
  "Biological Mathematics",
  "Ordinary Differential Equation Ontology",
  "Dynamical Systems Ontologies",
  "Stochastic Process Ontologies",
  "Graph Theory Ontologies",
  "Network Biology Ontologies",
  "Optimisation Ontologies",
  "Computational Mathematics Ontologies",
  "Parameter Estimation Ontologies"
 ]
 ROUTE_RULES = [
  "Systems Biology Mathematics -> Mathematical Biology for parent semantics",
  "Systems Biology Mathematics -> ODE Ontology for deterministic reaction dynamics",
  "Systems Biology Mathematics -> Dynamical Systems for stability and bifurcation",
  "Systems Biology Mathematics -> Stochastic Processes for molecular noise",
  "Systems Biology Mathematics -> Graph Theory and Network Biology for interaction structure",
  "Systems Biology Mathematics -> Optimisation for flux balance and calibration",
  "Systems Biology Mathematics -> Computational Mathematics for numerical simulation",
  "Systems Biology Mathematics -> Parameter Estimation for model inference"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible network types, kinetic models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve biological-network semantics.",
  "Preserve stoichiometry-versus-kinetics distinction.",
  "Preserve deterministic-versus-stochastic distinction.",
  "Preserve topology-versus-dynamics distinction.",
  "Preserve sensitivity-versus-identifiability distinction.",
  "Preserve steady-state-versus-transient distinction.",
  "Preserve model-validation semantics.",
  "Preserve machine-readable systems-biology-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Systems biology mathematics models interacting biological components and networks as integrated dynamical systems."
 REACTION_SUMMARY = "Reaction-network models combine stoichiometry with kinetic laws to describe biochemical dynamics."
 REGULATION_SUMMARY = "Gene-regulatory and signalling models represent activation, repression, feedback and information flow."
 METABOLIC_SUMMARY = "Metabolic models represent reaction fluxes, mass balance and constraint-based steady states."
 DYNAMICS_SUMMARY = "Feedback, stability, bifurcation, oscillation and multistability determine systems-level behaviour."
 STOCHASTIC_SUMMARY = "Chemical master equations and stochastic simulation represent molecular noise and single-cell variability."
 INFERENCE_SUMMARY = "Parameter estimation requires sensitivity, structural identifiability and practical identifiability analysis."
}

FLOW {
 INPUT
 -> detect_systems_biology_mathematics_context
 -> identify_biological_components_interactions_state_variables_parameters_scale_and_data
 -> construct_reaction_regulatory_signalling_metabolic_or_interaction_network
 -> choose_deterministic_stochastic_or_constraint_based_model
 -> formulate_stoichiometry_kinetics_equations_and_initial_conditions
 -> solve_or_simulate_system_dynamics
 -> identify_steady_states_fluxes_feedback_loops_and_network_motifs
 -> analyse_stability_oscillation_bistability_and_bifurcation
 -> estimate_parameters_and_assess_identifiability
 -> perform_sensitivity_uncertainty_and_model_reduction_analysis
 -> if_stochastic_simulate_noise_and_distributional_behaviour
 -> validate_against_biological_data_and_constraints
 -> route_to_mathematical_biology_ODE_dynamical_systems_stochastic_processes_graph_theory_network_biology_optimisation_computational_mathematics_or_parameter_estimation_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Systems Biology Mathematics Ontology defines mathematical analysis of interacting biological systems. It supports biochemical reaction networks, stoichiometry, mass-action, Michaelis-Menten and Hill kinetics, gene-regulatory, protein-interaction and signalling networks, metabolic networks and flux balance analysis, homeostasis, feedback and feed-forward motifs, oscillations, biological switches, bistability and multistability, Jacobian stability and bifurcation, parameter estimation, structural and practical identifiability, sensitivity, model reduction, timescale separation, stochastic systems biology, chemical master equations, Gillespie simulation, intrinsic and extrinsic noise, single-cell variability, computational systems biology and model validation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-SYSTEMS-BIOLOGY-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW