Biological Systems Analysis Ontology

flow "BiologicalSystemsAnalysisOntology" v1.0

# BIOLOGICAL SYSTEMS ANALYSIS ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/scientific-applications/biological-systems-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Scientific Applications"
ENTITY := "Biological Systems Analysis"

BIOLOGICAL_SYSTEMS_ANALYSIS := {
 type: AppliedBiomathematicalAnalysisFramework,
 domain: ScientificApplications,
 purpose: "Analyse biological systems using calculus, differential equations, dynamical systems, transport models, probability, optimisation and computational methods across molecular, cellular, organismal and population scales."
}

CONCEPTS := {
 BIOLOGICAL_SYSTEM,
 STATE_VARIABLE,
 POPULATION,
 CELL,
 CONCENTRATION,
 REACTION_RATE,
 GROWTH_RATE,
 DIFFERENTIAL_EQUATION,
 DYNAMICAL_SYSTEM,
 HOMEOSTASIS,
 FEEDBACK,
 STABILITY,
 REACTION_NETWORK,
 DIFFUSION,
 REACTION_DIFFUSION,
 PHARMACOKINETICS,
 EPIDEMIOLOGY,
 ECOLOGY,
 SYSTEMS_BIOLOGY
}

# ---------- BIOLOGICAL SYSTEM ----------
BIOLOGICAL_SYSTEM
 -> identify(
 ORGANISATION_LEVEL,
 STATE_VARIABLES,
 PARAMETERS,
 INTERACTIONS,
 INPUTS,
 OUTPUTS
)

levels:
 MOLECULAR
 CELLULAR
 TISSUE
 ORGAN
 ORGANISM
 POPULATION
 ECOSYSTEM

# ---------- STATE ----------
state:

x(t)
 :=
(
 x_1(t),
 ...,
 x_n(t)
)

x_i
 -> biological_quantity

examples:
 POPULATION_SIZE
 CELL_COUNT
 MOLECULE_CONCENTRATION
 BIOMASS
 MEMBRANE_POTENTIAL
 HORMONE_LEVEL

# ---------- DYNAMICAL MODEL ----------
dx/dt
 =
f(
 x,
 theta,
 u,
 t
)

theta
 -> MODEL_PARAMETERS

u
 -> EXTERNAL_INPUT

DYNAMICAL_SYSTEM
 -> biological_state_evolution

# ---------- GROWTH RATE ----------
population:

N(t)

absolute growth:

dN/dt

per-capita growth:

1/N
*
dN/dt

GROWTH_RATE
 -> local_population_change

# ---------- EXPONENTIAL GROWTH ----------
dN/dt
 =
r*N

solution:

N(t)
 =
N_0*exp(r*t)

r
 -> intrinsic_growth_rate

EXPONENTIAL_MODEL
 -> unlimited_growth_approximation

# ---------- LOGISTIC GROWTH ----------
dN/dt
 =
r*N
*
(
1-N/K
)

K
 -> CARRYING_CAPACITY

equilibria:

N_star
 =
0

N_star
 =
K

LOGISTIC_MODEL
 -> density_dependent_growth

# ---------- PREDATOR-PREY ----------
Lotka-Volterra:

dx/dt
 =
alpha*x
-
beta*x*y

dy/dt
 =
delta*x*y
-
gamma*y

x
 -> PREY

y
 -> PREDATOR

INTERACTION_TERM
 -> coupled_population_dynamics

# ---------- COMPETITION ----------
two species:

dx/dt
 =
r_1*x
*
(
1
-
(x+alpha_12*y)/K_1
)

dy/dt
 =
r_2*y
*
(
1
-
(y+alpha_21*x)/K_2
)

COMPETITION_COEFFICIENT
 -> interspecies_effect

ECOLOGICAL_OUTCOME
 -> coexistence_or_exclusion
    depending_on(parameters_and_initial_conditions)

# ---------- CHEMICAL / BIOCHEMICAL KINETICS ----------
reaction:

A+B
 ->
C

mass action rate:

v
 =
k*[A]*[B]

REACTION_RATE
 -> changes(CONCENTRATIONS)

stoichiometric system:

dx/dt
 =
S*v(x)

S
 -> STOICHIOMETRIC_MATRIX

# ---------- MICHAELIS-MENTEN ----------
enzyme reaction rate:

v
 =
V_max*[S]
/
(
K_m+[S]
)

MICHAELIS_MENTEN
 -> saturating_kinetic_model

validity
 -> depends_on(QUASI_STEADY_AND_MODEL_ASSUMPTIONS)

# ---------- HILL FUNCTION ----------
response:

R(x)
 =
x^n
/
(
K^n+x^n
)

n
 -> HILL_COEFFICIENT

HILL_FUNCTION
 -> cooperative_or_sigmoidal_response_model

# ---------- GENE REGULATION ----------
gene expression:

dm/dt
 =
transcription
-
degradation

dp/dt
 =
translation
-
degradation

REGULATORY_FUNCTION
 -> may_use(HILL_KINETICS)

GENE_NETWORK
 -> nonlinear_dynamical_system

# ---------- FEEDBACK ----------
POSITIVE_FEEDBACK
 -> reinforces(STATE_CHANGE)

NEGATIVE_FEEDBACK
 -> opposes(DEVIATION)

HOMEOSTASIS
 -> often_supported_by(NEGATIVE_FEEDBACK)

FEEDBACK
 -> changes(STABILITY_AND_RESPONSE)

# ---------- EQUILIBRIUM ----------
equilibrium:

f(x_star)
 =
0

BIOLOGICAL_STEADY_STATE
 -> constant_state_under_model

equilibrium
 -/> necessarily(healthy_or_desirable_state)

# ---------- LINEAR STABILITY ----------
Jacobian:

J
 :=
Df(x_star)

eigenvalues
 -> local_stability

Re(lambda_i)<0
 for all i

-> locally_asymptotically_stable

positive real part
 -> instability

NONHYPERBOLIC_CASE
 -> requires(FURTHER_ANALYSIS)

# ---------- HOMEOSTASIS ----------
regulated variable:

y

target region:

y_star

perturbation
 -> feedback_response
 -> return_toward_operating_state

HOMEOSTASIS
 -> dynamical_regulation_property

HOMEOSTASIS
 -/> identical_to(static_equilibrium)

# ---------- OSCILLATION ----------
biological systems
 -> may exhibit(PERIODIC_DYNAMICS)

examples:
 CIRCADIAN_RHYTHM
 CELL_CYCLE
 CALCIUM_OSCILLATION
 NEURAL_RHYTHM

LIMIT_CYCLE
 -> model_of(SUSTAINED_BIOLOGICAL_OSCILLATION)

# ---------- BIFURCATION ----------
parameter change
 -> qualitative_change_in(BIOLOGICAL_DYNAMICS)

examples:
 equilibrium_to_oscillation
 extinction_to_persistence
 monostability_to_bistability

BIFURCATION_ANALYSIS
 -> identifies(CRITICAL_PARAMETER_VALUES)

# ---------- SPATIAL DIFFUSION ----------
concentration:

c(x,t)

diffusion:

partial_c/partial_t
 =
D*LAPLACIAN(c)

D
 -> DIFFUSION_COEFFICIENT

DIFFUSION
 -> spatial_spreading_of(BIOLOGICAL_MATERIAL)

# ---------- REACTION-DIFFUSION ----------
partial_u/partial_t
 =
D_u*LAPLACIAN(u)
+
f(u,v)

partial_v/partial_t
 =
D_v*LAPLACIAN(v)
+
g(u,v)

REACTION_DIFFUSION
 -> local_kinetics_plus_spatial_transport

# ---------- TURING PATTERN ----------
homogeneous equilibrium
 -> stable_without_diffusion

but
 -> unstable_with_differential_diffusion

under suitable conditions

TURING_INSTABILITY
 -> spatial_pattern_formation

# ---------- CHEMOTAXIS ----------
cell density:

n(x,t)

chemical:

c(x,t)

cell flux
 -> diffusion
 + directed_motion_along(chemical_gradient)

CHEMOTAXIS
 -> gradient-guided_biological_transport

# ---------- MEMBRANE TRANSPORT ----------
flux:

J
 =
-P
*
(
 C_inside-C_outside
)

in simple permeability model

more detailed:
 FICK
 NERNST_PLANCK
 ACTIVE_TRANSPORT

MEMBRANE_TRANSPORT
 -> movement_across(BIOLOGICAL_BOUNDARY)

# ---------- ELECTRODIFFUSION ----------
ionic flux:

J_i
 =
-D_i
*
(
 grad(c_i)
 +
(z_i*F/(R*T))*c_i*grad(phi)
)

NERNST_PLANCK
 -> diffusion_plus_electric_drift

ION_TRANSPORT
 -> coupled_to(ELECTRIC_POTENTIAL)

# ---------- MEMBRANE POTENTIAL ----------
Nernst potential:

E_i
 =
(R*T/(z_i*F))
*
ln(
 c_out/c_in
)

under idealised equilibrium assumptions

MEMBRANE_VOLTAGE
 -> electrochemical_gradient_relation

# ---------- NEURAL DYNAMICS ----------
membrane equation:

C_m*dV/dt
 =
-I_ion
+
I_ext

HODGKIN_HUXLEY
 -> conductance_based_neuron_model

ION_CHANNEL_GATING
 -> nonlinear_state_dynamics

# ---------- PHARMACOKINETICS ----------
one-compartment model:

dC/dt
 =
-(CL/V)*C
+
INPUT/V

CL
 -> CLEARANCE

V
 -> DISTRIBUTION_VOLUME

PHARMACOKINETICS
 -> drug_concentration_dynamics

# ---------- EXPONENTIAL ELIMINATION ----------
after bolus input and first-order elimination:

C(t)
 =
C_0*exp(-k*t)

half-life:

t_half
 =
ln(2)/k

ELIMINATION_RATE
 -> determines(DRUG_DECAY)

# ---------- MULTICOMPARTMENT ----------
central
 <->
peripheral

coupled ODEs
 -> DISTRIBUTION_AND_ELIMINATION

COMPARTMENT_MODEL
 -> reduced_physiological_representation

# ---------- EPIDEMIOLOGY ----------
SIR model:

dS/dt
 =
-beta*S*I/N

dI/dt
 =
beta*S*I/N
-
gamma*I

dR/dt
 =
gamma*I

S
 -> susceptible

I
 -> infectious

R
 -> removed_or_recovered_context

# ---------- BASIC REPRODUCTION NUMBER ----------
for standard SIR assumptions:

R_0
 =
beta/gamma

when parameters defined consistently

R_0
 -> expected_secondary_infections_in_fully_susceptible_context

R_0
 -/> universal_constant_across_models_or_populations

# ---------- POPULATION GENETICS ----------
allele frequency:

p

selection
 mutation
 migration
 drift

-> change(p)

POPULATION_GENETICS
 -> deterministic_and_stochastic_models

# ---------- STOCHASTIC BIOLOGY ----------
small populations
 -> random_fluctuations_significant

model:

dX
 =
f(X,t)dt
+
g(X,t)dW_t

STOCHASTIC_DIFFERENTIAL_EQUATION
 -> biological_noise_model

# ---------- MASTER EQUATION ----------
probability:

P(n,t)

chemical master equation
 -> state_transition_probability_dynamics

GILLESPIE_ALGORITHM
 -> stochastic_reaction_simulation

DETERMINISTIC_ODE
 -/> always_valid_at(LOW_COPY_NUMBER)

# ---------- PARAMETER ESTIMATION ----------
observations:

y_i

model:

y_hat_i(theta)

objective:

J(theta)
 =
sum_i(
 y_i-y_hat_i(theta)
)^2

PARAMETER_ESTIMATION
 -> infer(BIOLOGICAL_RATES_AND_CONSTANTS)

# ---------- IDENTIFIABILITY ----------
STRUCTURAL_IDENTIFIABILITY
 -> uniqueness_under_ideal_data_and_model

PRACTICAL_IDENTIFIABILITY
 -> estimability_under(FINITE_NOISY_DATA)

SENSITIVITY
 -> informs(PARAMETER_IDENTIFIABILITY)

# ---------- UNCERTAINTY ----------
biological parameters
 -> uncertain_and_variable

UNCERTAINTY_ANALYSIS
 -> propagate(PARAMETER_AND_MEASUREMENT_UNCERTAINTY)

methods:
 MONTE_CARLO
 BAYESIAN_INFERENCE
 SENSITIVITY_ANALYSIS
 BOOTSTRAP

# ---------- MULTISCALE ----------
molecular processes
 -> cellular behaviour

cellular behaviour
 -> tissue dynamics

tissue dynamics
 -> organismal_or_population_effects

MULTISCALE_MODEL
 -> couples(DIFFERENT_SPATIAL_AND_TEMPORAL_SCALES)

# ---------- NETWORK ANALYSIS ----------
biological network:

nodes
 -> genes_proteins_metabolites_species

edges
 -> interactions

NETWORK_TOPOLOGY
 -> complements(DYNAMICAL_ANALYSIS)

GRAPH_STRUCTURE
 -/> determines_dynamics_without(PARAMETERS_AND_KINETICS)

# ---------- CONTROL ----------
biological intervention:

u(t)

system:

dx/dt
 =
f(x,u,t)

OPTIMAL_CONTROL
 -> choose(u)
 -> achieve(BIOLOGICAL_OBJECTIVE)

applications:
 TREATMENT_SCHEDULING
 BIOREACTOR_CONTROL
 ECOLOGICAL_MANAGEMENT

# ---------- NUMERICAL METHODS ----------
biological models
 -> solved_by(
 ODE_SOLVER,
 PDE_SOLVER,
 STOCHASTIC_SIMULATION,
 AGENT_BASED_SIMULATION,
 OPTIMISATION
)

NUMERICAL_METHOD
 -> must_match(MODEL_SCALE_STIFFNESS_AND_STOCHASTICITY)

# ---------- VALIDATION ----------
model prediction
 -> compare_with(BIOLOGICAL_DATA)

VALIDATION
 -> assess(MODEL_SCOPE_AND_PREDICTIVE_UTILITY)

parameter fit
 -/> sufficient_validation

BIOLOGICAL_VARIABILITY
 -> part_of(VALIDATION_CONTEXT)

# ---------- APPLICATIONS ----------
BIOLOGICAL_SYSTEMS_ANALYSIS
 -> SYSTEMS_BIOLOGY
 -> ECOLOGY
 -> EPIDEMIOLOGY
 -> PHYSIOLOGY
 -> NEUROSCIENCE
 -> PHARMACOLOGY
 -> GENETICS
 -> BIOENGINEERING
 -> CELL_BIOLOGY
 -> BIOMEDICAL_MODELLING

# ---------- RELATION GRAPH ----------
BIOLOGICAL_STATE
 -> evolves_by(DYNAMICAL_MODEL)

REACTION_NETWORK
 -> generates(KINETIC_EQUATIONS)

FEEDBACK
 -> controls(STABILITY_AND_HOMEOSTASIS)

DIFFUSION
 -> generates(SPATIAL_TRANSPORT)

REACTION_DIFFUSION
 -> generates(PATTERNS)

DATA
 -> constrains(PARAMETERS)

SENSITIVITY
 -> informs(IDENTIFIABILITY)

UNCERTAINTY
 -> qualifies(PREDICTION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "biological systems analysis" ~= "mathematical analysis of interacting biological processes across time and space",
 "homeostasis" ~= "dynamic regulation maintaining biological variables within functional ranges",
 "reaction network" ~= "interacting biochemical transformations represented by species and reaction rates",
 "reaction-diffusion system" ~= "biological field model combining local reactions with spatial diffusion",
 "parameter identifiability" ~= "degree to which model parameters can be uniquely or reliably inferred from observations",
 "multiscale model" ~= "model coupling biological processes operating at different spatial or temporal scales"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 biological_models_are_simplified_representations_not_complete_organisms: true,
 equilibrium_does_not_automatically_mean_homeostasis_or_health: true,
 deterministic_models_may_fail_for_small_noisy_populations: true,
 parameter_fit_does_not_guarantee_identifiability_or_predictive_validity: true,
 biological_variability_and_measurement_uncertainty_must_be_preserved: true,
 model_parameters_and_R0_like_quantities_are_context_and_model_dependent: true,
 numerical_predictions_require_solver_convergence_and_data_validation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 population_kinetic_relation: 0.99,
 stability_feedback_relation: 0.99,
 spatial_transport_relation: 0.99,
 epidemiology_pharmacokinetic_relation: 0.99,
 identifiability_uncertainty_relation: 0.98,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(biological_scale_state_variables_parameters_interactions_and_data)
 -> choose(deterministic_stochastic_spatial_network_or_multiscale_model)
 -> formulate(conservation_growth_reaction_transport_and_feedback_relations)
 -> determine(equilibria_stability_oscillation_and_bifurcation_structure)
 -> analyse(spatial_patterns_transport_or_stochastic_effects_if_relevant)
 -> estimate(parameters_and_test_identifiability)
 -> quantify(sensitivity_uncertainty_and_biological_variability)
 -> solve(model_with_appropriate_numerical_method)
 -> compare(predictions_with_data)
 -> validate(model_scope_assumptions_and_predictive_limits)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed biological-systems-analysis semantic graph",
 primary_entity: BIOLOGICAL_SYSTEMS_ANALYSIS,
 preserve: [
  biological_scale,
  state_variables,
  parameters,
  interactions,
  initial_conditions,
  spatial_domain,
  stochasticity,
  observations,
  uncertainty
 ],
 infer_through: [
  population_dynamics,
  biochemical_kinetics,
  feedback,
  homeostasis,
  stability,
  reaction_diffusion,
  pharmacokinetics,
  epidemiology,
  identifiability,
  uncertainty_analysis
 ],
 avoid: [
  treating_model_as complete_biological_reality,
  equating_equilibrium_with healthy_homeostasis,
  ignoring_stochasticity_at low copy number,
  claiming_parameter_uniqueness_from fit alone,
  treating_context_specific_parameters_as universal constants,
  accepting_simulation_without biological_and_numerical validation
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(biological_systems_analysis_context)
 -> compare(existing_relations)
 -> preserve(scale_state_parameter_interaction_data_and_uncertainty_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + BIOLOGICAL_SYSTEMS_ANALYSIS
 + BIOLOGICAL_SYSTEM
 + STATE_VARIABLE
 + POPULATION
 + REACTION_RATE
 + DIFFERENTIAL_EQUATION
 + HOMEOSTASIS
 + FEEDBACK
 + STABILITY
 + REACTION_NETWORK
 + REACTION_DIFFUSION
 + PHARMACOKINETICS
 + EPIDEMIOLOGY
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> BIOLOGICAL_SYSTEMS_ANALYSIS
 -> SYSTEM_SCALE_DATA_IDENTIFICATION
 -> MODEL_CLASSIFICATION
 -> BIOLOGICAL_RELATION_FORMULATION
 -> DYNAMICAL_STABILITY_ANALYSIS
 -> SPATIAL_STOCHASTIC_ANALYSIS
 -> PARAMETER_IDENTIFIABILITY_ANALYSIS
 -> SENSITIVITY_UNCERTAINTY_ANALYSIS
 -> NUMERICAL_SOLUTION
 -> DATA_COMPARISON
 -> MODEL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW