Prostate cancer driven by activated androgen receptors
The Jain model of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
- Name:
prostate_androgen_receptor - Field:
biology/oncology - Formalism: ordinary differential equations
- References: Padmanabhan, Meskin, and Al Moustafa 2021; Jain et al. 2011
Factory
@register_model(
field='biology/oncology',
tags=('hormone therapy', 'ode', 'prostate cancer', 'androgen receptor', 'PSA'),
references=(BOOK, 'Jain et al. 2011'),
defaults={'until': 26280.0, 'dt': 168.0, 'solver': ScipyODESolver(method='LSODA')},
)
def prostate_androgen_receptor(
N: float = 1.5,
AD: float = 1.0,
AI: float = 0.0,
*,
UT: float = 40.0,
UD: float = 40.0,
m1: float = 1e-4,
age: float = 60.0,
) -> System:
"""Return the Jain model of prostate epithelium and cancer driven by activated androgen receptors, with PSA.
The growth and death rates of the normal, androgen-dependent, and androgen-independent cells follow the receptors
activated by testosterone, `UT`, and dihydrotestosterone, `UD`, which are parameters, so androgen ablation is a
lower activation. The normal cells fill a capacity that grows with the age-dependent prostate volume, and the PSA
the cells produce in the tissue leaks into the serum through the normal epithelium and, faster, through the
tumour. Cells are in billions, receptors in nM, PSA in ng/mL, and time in hours, the age in years. The book
leaves several parts open, filled as follows. The half saturations `psi2`, `phi2`, and `psi4` of the cancer cells
are those of the normal cells. The death rate is `psi3_i / (psi4 + UD + UT)`. One activation `UT`, `UD` holds
for all cell types, and so does `sigma`. The independent cells gain the part `m1` of the divisions of the
dependent cells. The negligible interaction `c1` is 0. The capacity of the normal cells is the prostate volume
over the volume of a billion cells, and the PSA leak rate per million cells becomes `0.3125` per billion.
Args:
N:
The initial normal epithelial cells.
AD:
The initial androgen-dependent cells.
AI:
The initial androgen-independent cells.
UT:
The concentration of receptors activated by testosterone.
UD:
The concentration of receptors activated by dihydrotestosterone.
m1:
The fraction of the divisions of dependent cells that yields independent cells.
age:
The age of the patient at the start.
Returns:
system:
The model, with the cells `prostate.N`, `prostate.AD`, and `prostate.AI`, and the PSA `prostate.UIP` in
the tissue and `prostate.UP` in the serum.
Examples:
>>> from skcomplex.models.biology.oncology import prostate_androgen_receptor
>>> prostate_androgen_receptor(UT=5.0).elements[0].UT
5.0
"""
prostate = AndrogenReceptorProstateElement('prostate', N=N, AD=AD, AI=AI, UT=UT, UD=UD, m1=m1, age=age)
interactions = [
RateInteraction(
'normal_growth',
target='prostate',
state='N',
rate=lambda s: _receptor_growth(s.target, s.target.psi1_n, s.target.phi1_n)
* s.target.N
* (
1
- (s.target.N + s.target.c1 * (s.target.AD + s.target.AI))
/ (_prostate_volume(s.target, s.t) / s.target.Vc)
),
),
ReactionInteraction(
'normal_death',
target='prostate',
reactants={'N': 1},
propensity=lambda s: _receptor_death(s.target, s.target.psi3_n) * s.target.N,
),
RateInteraction(
'dependent_division',
target='prostate',
state='AD',
rate=lambda s: _dependent_division(s.target),
),
ReactionInteraction(
'mutation',
target='prostate',
reactants={'AD': 1},
products={'AI': 1},
propensity=lambda s: s.target.m1 * _dependent_division(s.target),
),
ReactionInteraction(
'dependent_death',
target='prostate',
reactants={'AD': 1},
propensity=lambda s: _receptor_death(s.target, s.target.psi3_d) * s.target.AD,
),
RateInteraction(
'independent_growth',
target='prostate',
state='AI',
rate=lambda s: _receptor_growth(s.target, s.target.psi1_i, s.target.phi1_i)
* s.target.AI
* (1 - (s.target.c1 * s.target.N + s.target.AD + s.target.AI) / s.target.b3),
),
ReactionInteraction(
'independent_death',
target='prostate',
reactants={'AI': 1},
propensity=lambda s: _receptor_death(s.target, s.target.psi3_i) * s.target.AI,
),
RateInteraction(
'psa_production',
target='prostate',
state='UIP',
rate=lambda s: (s.target.UT + s.target.UD + s.target.sigma)
* (s.target.beta_n * s.target.N + s.target.beta_d * s.target.AD + s.target.beta_i * s.target.AI),
),
ReactionInteraction(
'normal_leak',
target='prostate',
reactants={'UIP': 1},
products={'UP': 1},
propensity=lambda s: s.target.du1
* s.target.UIP
* s.target.N
/ (_prostate_volume(s.target, s.t) / s.target.Vc),
),
ReactionInteraction(
'cancer_leak',
target='prostate',
reactants={'UIP': 1},
products={'UP': 1},
propensity=lambda s: s.target.gamma_p
* s.target.UIP
* (s.target.AD + s.target.AI) ** 2
/ (s.target.KP + s.target.AD + s.target.AI),
),
ReactionInteraction(
'intracellular_clearance',
target='prostate',
reactants={'UIP': 1},
propensity=lambda s: s.target.du2 * s.target.UIP,
),
ReactionInteraction(
'serum_clearance',
target='prostate',
reactants={'UP': 1},
propensity=lambda s: s.target.du3 * s.target.UP,
),
]
return System(elements=[prostate], interactions=interactions)
Components
class AndrogenReceptorProstateElement(Element):
"""Normal epithelium and prostate cancer cells driven by activated androgen receptors, with tissue and serum PSA.
The normal cells follow `dN/dt = r1 N (1 - (N + c1 (AD + AI)) / b1) - d1 N`, the dependent cells
`dAD/dt = r2 AD (1 - m1)(1 - L) - d2 AD`, and the independent cells `dAI/dt = r3 AI (1 - (c1 N + AD + AI) / b3)
- d3 AI + m1 r2 AD (1 - L)`, where `L = (c1 N + AD + AI) / b2`. The rates are
`r_i = psi1_i UD^2 / (psi2^2 + UD^2) + phi1_i UT^2 / (phi2^2 + UT^2)` and `d_i = psi3_i / (psi4 + UD + UT)` for
the receptors activated by DHT, `UD`, and by testosterone, `UT`. The tissue PSA follows
`dUIP/dt = -du1 UIP Nf - gamma_p UIP (AD + AI)^2 / (KP + AD + AI) - du2 UIP + r4`, with the production
`r4 = sum_i beta_i (UT + UD + sigma) X_i` over the cell types, and the serum PSA gains the leaks and is cleared
at `du3`. The capacity of the normal cells `b1` and their share `Nf = N / b1` follow the prostate volume
`v1 a^phi / (v2^phi + a^phi) + v3` at the age `a`, divided by the volume `Vc` of a billion cells.
The element holds the states and parameters and has no laws of its own: the interactions of
`prostate_androgen_receptor` read its parameters.
Args:
label:
The label of the prostate.
visual:
How the element is drawn, or `None` for `default_visual_`.
N:
The initial normal epithelial cells.
AD:
The initial androgen-dependent cells.
AI:
The initial androgen-independent cells.
UIP:
The initial tissue PSA.
UP:
The initial serum PSA.
UT:
The concentration of receptors activated by testosterone.
UD:
The concentration of receptors activated by dihydrotestosterone.
m1:
The fraction of the divisions of dependent cells that yields independent cells.
b2:
The carrying capacity of the dependent cells.
b3:
The carrying capacity of the independent cells.
c1:
The interaction between normal and cancer cells.
psi1_n:
The maximum growth of the normal cells driven by DHT.
psi1_d:
The maximum growth of the dependent cells driven by DHT.
psi1_i:
The maximum growth of the independent cells driven by DHT.
phi1_n:
The maximum growth of the normal cells driven by testosterone.
phi1_d:
The maximum growth of the dependent cells driven by testosterone.
phi1_i:
The maximum growth of the independent cells driven by testosterone.
psi2:
The DHT-activated receptors at which the growth is half way.
phi2:
The testosterone-activated receptors at which the growth is half way.
psi3_n:
The death of the normal cells.
psi3_d:
The death of the dependent cells.
psi3_i:
The death of the independent cells.
psi4:
The activated receptors at which the death is halved.
Vc:
The volume of a billion cells.
v1:
The prostate volume that ageing adds.
v2:
The age at which half of that volume is added.
v3:
The prostate volume of a young man.
phi:
The steepness of the growth of the prostate with age.
age:
The age of the patient at the start.
du1:
The rate at which tissue PSA leaks through normal epithelium.
du2:
The decay rate of tissue PSA.
du3:
The clearance rate of serum PSA.
gamma_p:
The rate at which tissue PSA leaks through cancer cells.
KP:
The cancer cells at which the leak per cell is half its maximum.
beta_n:
The PSA production of the normal cells.
beta_d:
The PSA production of the dependent cells.
beta_i:
The PSA production of the independent cells.
sigma:
The PSA production independent of the receptors.
Examples:
>>> from skcomplex.models.biology.oncology import AndrogenReceptorProstateElement
>>> AndrogenReceptorProstateElement('prostate').KP
800.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
N: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 1.5,
AD: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
AI: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
UIP: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
UP: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
UT: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 40.0,
UD: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 40.0,
m1: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.0001,
b2: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 22.4,
b3: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 22.4,
c1: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
psi1_n: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.0001202,
psi1_d: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.001048,
psi1_i: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 9.8e-05,
phi1_n: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.0001281,
phi1_d: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.00111,
phi1_i: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.000279,
psi2: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 12.355,
phi2: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 49.587,
psi3_n: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.008,
psi3_d: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.15,
psi3_i: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.01,
psi4: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 62.162,
Vc: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 5.56,
v1: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 18.397,
v2: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 63.753,
v3: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 28.4,
phi: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 12.0,
age: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 60.0,
du1: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.001515,
du2: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.05635,
du3: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 5.776,
gamma_p: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.3125,
KP: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 800.0,
beta_n: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.0022622,
beta_d: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.002,
beta_i: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.003,
sigma: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 10.0,
) -> None:
super().__init__(label, visual=visual)
self.N = N
self.AD = AD
self.AI = AI
self.UIP = UIP
self.UP = UP
self.UT = UT
self.UD = UD
self.m1 = m1
self.b2 = b2
self.b3 = b3
self.c1 = c1
self.psi1_n = psi1_n
self.psi1_d = psi1_d
self.psi1_i = psi1_i
self.phi1_n = phi1_n
self.phi1_d = phi1_d
self.phi1_i = phi1_i
self.psi2 = psi2
self.phi2 = phi2
self.psi3_n = psi3_n
self.psi3_d = psi3_d
self.psi3_i = psi3_i
self.psi4 = psi4
self.Vc = Vc
self.v1 = v1
self.v2 = v2
self.v3 = v3
self.phi = phi
self.age = age
self.du1 = du1
self.du2 = du2
self.du3 = du3
self.gamma_p = gamma_p
self.KP = KP
self.beta_n = beta_n
self.beta_d = beta_d
self.beta_i = beta_i
self.sigma = sigma
class RateInteraction(Interaction):
"""Rate `rate(ns)` and noise `noise(ns) dW` added to the highest time derivative of a state of a target.
On a state of order 1 the rate is its first derivative, and on a state of order `k` it adds to the `k`-th
derivative, so rates on one state superpose. With a divisor `d` the rate and the noise are divided by `d`, the name
of a state or parameter of the target, such as an inertia, a friction, or a capacitance, which backends can read.
A rate, a noise, or a divisor function takes the namespace of the interaction, with the namespaces `target` and
`sources` of the bound elements, the time `t`, and the symbols of the parameters a subclass declares. A number
stands for a constant, and a number on a vector state applies to every component. The noise is a Wiener process per
component and member by default, a process of a `WienerProcessSpec` to share or correlate it across states and
members, or a fractional Brownian motion of a Hurst index.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state the rate changes.
state:
The name of the state of the target that the rate changes.
rate:
A function of the namespace that returns the rate, or a number.
sources:
The paths of the elements or groups the rate and the noise read but do not change.
noise:
A function of the namespace that returns the amplitude of the noise, a number, or `None` for no noise.
divisor:
The name of the state or parameter of the target that divides the rate and the noise, a function of the
namespace of the target that returns the divisor, such as a moment of inertia `m l^2`, or `None`.
process:
The Wiener processes that drive the noise, or `None` for one independent process per component and member.
index:
The process of `process` that drives the noise, or `None` for every process, one per component of a vector
state.
convention:
The stochastic calculus of the noise, `'ito'`, `'stratonovich'`, or `'anti-ito'`.
hurst:
The Hurst index in `(0, 1)` of a fractional Brownian motion that drives the noise, a number or a function
of the namespace that returns a parameter, or `None` for a Wiener process.
conserved:
Whether the noise is the divergence of a random flux on the links of the lattice of a field, so it keeps the
total of the state.
Raises:
ValidationError:
The state is not a name, a function is neither callable nor a number, the convention is unknown, the Hurst
index is outside `(0, 1)`, or the noise options cannot combine.
Examples:
>>> from skcomplex.core.elements import CompartmentElement
>>> from skcomplex.core.interactions import RateInteraction
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> decay = RateInteraction('decay', target='stock', state='n', rate=lambda s: -0.5 * s.target.n)
>>> system = System(elements=[CompartmentElement('stock', n=8.0)], interactions=[decay])
>>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=2.0).trajectory_
>>> round(float(trajectory.values('stock.n')[-1]), 3)
2.943
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str = 'x',
rate: Callable[[SimpleNamespace], object] | float = 0.0,
sources: Annotated[Sequence[str], RoleSpec(variadic=True)] = (),
noise: Callable[[SimpleNamespace], object] | float | None = None,
divisor: str | Callable[[SimpleNamespace], object] | None = None,
process: Annotated[WienerProcessSpec | None, RoleSpec(noise=True)] = None,
index: int | None = None,
convention: str = 'ito',
hurst: Callable[[SimpleNamespace], object] | float | None = None,
conserved: bool = False,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.rate = rate
self.sources = sources
self.noise = noise
self.divisor = divisor
self.process = process
self.index = index
self.convention = convention
self.hurst = hurst
self.conserved = conserved
def _check_functions(self) -> None:
"""Raise an error for a state that is not a name, or a function that is neither callable nor a number."""
if not isinstance(self.state, str) or not self.state.isidentifier():
error_msg = (
f'{type(self).__name__} {self.label!r} changes the state {self.state!r}, which is not a name of a '
'state.'
)
raise ValidationError(error_msg)
for name in ('rate', 'noise'):
value = getattr(self, name)
if not (value is None or callable(value) or isinstance(value, int | float)):
error_msg = f'The {name} of {self.label!r} is {value!r}. Give a function of the namespace or a number.'
raise ValidationError(error_msg)
if not (
self.divisor is None
or callable(self.divisor)
or (isinstance(self.divisor, str) and self.divisor.isidentifier())
):
error_msg = (
f'The divisor of {self.label!r} is {self.divisor!r}. Give the name of a state or parameter, or a '
'function of the namespace of the target.'
)
raise ValidationError(error_msg)
def _check_noise(self) -> None:
"""Raise an error for an unknown convention, a Hurst index outside `(0, 1)`, or noise options that conflict."""
if self.convention not in CONVENTIONS:
error_msg = f'Unknown noise convention {self.convention!r}. Use one of {", ".join(CONVENTIONS)}.'
raise ValidationError(error_msg)
if not (
self.hurst is None or callable(self.hurst) or (isinstance(self.hurst, int | float) and 0 < self.hurst < 1)
):
error_msg = f'The Hurst index of {self.label!r} is {self.hurst!r}. Give a number in (0, 1) or a function.'
raise ValidationError(error_msg)
if self.index is not None and self.process is None:
error_msg = f'{self.label!r} picks the process {self.index!r} but binds no processes. Give `process`.'
raise ValidationError(error_msg)
options = [
name
for name, given in (('hurst', self.hurst), ('process', self.process), ('conserved', self.conserved))
if given
]
if len(options) > 1:
error_msg = f'The noise of {self.label!r} takes {" and ".join(options)}. Give at most one of them.'
raise ValidationError(error_msg)
def _check(self) -> None:
"""Raise an error for a state, functions, or noise options the interaction cannot take.
Raises:
ValidationError:
The type declares states, the state is not a name, a function is neither callable nor a number, the
convention is unknown, the Hurst index is outside `(0, 1)`, or the noise options cannot combine.
"""
super()._check()
self._check_functions()
self._check_noise()
def _evaluate(self, function: Callable[[SimpleNamespace], object] | float, namespace: SimpleNamespace) -> object:
"""Return the value of a function of the namespace, or a number as it is."""
return function(namespace) if callable(function) else function
def _broadcast(self, value: object, state: object) -> object:
"""Return a value with one component per component of a vector state, repeating a number."""
if isinstance(state, FreeModuleElement) and not isinstance(value, FreeModuleElement | list | tuple):
return vector(SR, [value] * len(state))
if isinstance(value, list | tuple):
return vector(SR, value)
return value
def _divide(self, value: object, namespace: SimpleNamespace) -> object:
"""Return a value divided by the divisor of the target, or the value without a divisor."""
if self.divisor is None:
return value
divisor = (
self.divisor(self._get_member(namespace, 'target'))
if callable(self.divisor)
else getattr(self._get_member(namespace, 'target'), self.divisor)
)
return operator.truediv(value, divisor)
def _get_rate(self, namespace: SimpleNamespace) -> object:
"""Return the rate before the divisor, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.rate, namespace)
def _get_noise(self, namespace: SimpleNamespace) -> object:
"""Return the noise amplitude before the divisor, the hook a subclass overrides, or `None` for no noise."""
return None if self.noise is None else self._evaluate(self.noise, namespace)
def _get_noise_law(self, state: object, amplitude: object, namespace: SimpleNamespace) -> BasePart:
"""Return the noise law of an amplitude, by the process, the convention, and the Hurst index."""
if self.hurst is not None:
hurst = self._evaluate(self.hurst, namespace)
return FractionalNoiseLaw(state, amplitude, hurst=hurst, convention=self.convention)
process = None
if self.process is not None:
handle = self._get_member(namespace, 'process').dW
process = handle if self.index is None else handle[self.index]
return NoiseLaw(state, amplitude, convention=self.convention, process=process, conserved=self.conserved)
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the rate law of the state and its noise law, both divided by the divisor.
Args:
namespace:
The namespace of the symbols the laws can use, with the namespaces of the target and the sources.
Returns:
laws:
The rate law, and the noise law when the interaction has noise.
"""
state = getattr(self._get_member(namespace, 'target'), self.state)
rate = self._divide(self._broadcast(self._get_rate(namespace), state), namespace)
laws: list[BasePart] = [RateLaw(state, rate)]
noise = self._get_noise(namespace)
if noise is not None:
laws.append(self._get_noise_law(state, self._divide(noise, namespace), namespace))
return laws
class ReactionInteraction(JumpInteraction):
"""Reaction that moves units from its reactants to its products at a mass-action or a given propensity.
A reactant, a product, or a modifier is a pair of the path of an element, or of a member of a group such as
`cells[0]`, and the name of its state that holds the units, mapped to its stoichiometric coefficient. A coefficient
is a positive number, or a function of the namespace of the reaction that returns an expression of its parameters,
such as `lambda s: 1 / s.target.epsilon`. So `S + I -> 2 I` has the reactants `{('S', 'n'): 1, ('I', 'n'): 1}` and
the products `{('I', 'n'): 2}`, and two states of one element are `('cell', 'u')` and `('cell', 'v')`. A species may
also be the name of a state of the `target` alone, such as `{'u': 1, 'v': 2}`. A target that is a group or a field
reacts in every member, with the species of every member its own states, as the reactions at every site of a
reaction-diffusion field.
A subclass may bind its species under roles of its own names, such as `susceptible` and `infected`: it returns its
entries from `_get_reactants`, `_get_products`, and `_get_modifiers`, pairs of the path a role binds and a state
name, and the reaction reads every pair from the namespace of the role that binds its path.
The propensity is `k` times the product of every reactant and modifier state to the power of its coefficient, or the
expression the `propensity` function returns. The reaction changes every state by its coefficient among the products
minus its coefficient among the reactants, so it conserves the total of the states whose changes sum to zero, and
modifiers enter the propensity but do not change.
In the mode `'rate'` the reaction is a rate equation. In the mode `'jump'` it is the random event of a
`JumpInteraction` at the propensity that moves the units of the coefficients. In the mode `'langevin'` it is the
chemical Langevin equation, the rate equation with one Wiener process of amplitude the square root of the propensity
driving every state the reaction changes.
In the `'agent'` form the target is an agent group and every member reacts on its own states, the species named by
state names alone, in every step from the states at its start. The propensity of a member is `k` times the product
of its reactant and modifier states to the power of their coefficients, or the number `propensity(agent, model)`
returns, and a coefficient that is a function is the number `coefficient(agent, model)` returns. In the mode
`'rate'` the member adds the changes times its propensity, one unit step of the rate equation, and in the mode
`'jump'` it adds the changes with the probability `1 - exp(-propensity)` that an event fires in the step. The
members react at once from the states at the start of the rule under the schedule `'synchronous'`, one after the
other in their order or in a new random order every step under `'sequential'` or `'random-sequential'`, or each in
its own activation under `'member'`.
Args:
label:
The label of the reaction, unique among the parts of its system.
target:
The path of the element, group, or field whose states the species named by a state alone are, or `None`.
reactants:
The coefficients of the reactants, or `None` for a source.
products:
The coefficients of the products, or `None` for a sink.
k:
The rate constant.
propensity:
A function of the namespace of the reaction that returns the propensity, or `None` for mass action. The
namespace holds the tuples `reactants`, `products`, and `modifiers` of the namespaces of the elements their
pairs bind, in the order of their entries, `target`, the namespace of the target or of one of its members,
and the symbol `k`. In the agent form, a function `propensity(agent, model)` that returns a number.
modifiers:
The coefficients of the modifiers, whose states the propensity reads but the reaction does not change, or
`None`.
mode:
The form of the laws, `'rate'`, `'jump'`, or `'langevin'`.
form:
`'symbolic'` for the laws of the states, or `'agent'` for a rule in which every member of an agent group
reacts on its own states.
phase:
The phase of the step the agent form runs in, or `None` for the default phase.
schedule:
The order of the members in the agent form, `'synchronous'`, `'sequential'`, `'random-sequential'`, or
`'member'`.
Raises:
ValidationError:
The mode, the form, or the schedule is unknown, the symbolic form is not synchronous, the reaction has
neither reactants nor products, an entry is neither a pair of a path and a state name nor a state name of a
given target, a coefficient is neither a positive number nor a function, or the agent form has no target,
has an entry that is a pair, or has the mode `'langevin'`.
Examples:
>>> from skcomplex.core.elements import CompartmentElement
>>> from skcomplex.core.interactions import ReactionInteraction
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> infection = ReactionInteraction(
... 'infection', reactants={('S', 'n'): 1, ('I', 'n'): 1}, products={('I', 'n'): 2}, k=0.0005
... )
>>> recovery = ReactionInteraction('recovery', reactants={('I', 'n'): 1}, products={('R', 'n'): 1}, k=0.1)
>>> system = System(
... elements=[CompartmentElement('S', n=990), CompartmentElement('I', n=10), CompartmentElement('R')],
... interactions=[infection, recovery],
... )
>>> states = Simulation(system, dt=1.0).check().prepare().run(until=50).trajectory_.states
>>> round(float(states[-1].sum()), 6)
1000.0
The decay `n -> 0` at every site of a field, at the rate `k n` of the site:
>>> import numpy as np
>>> from skcomplex.core.elements import FieldElement
>>> from skcomplex.core.spaces import LatticeSpace
>>> decay = ReactionInteraction('decay', target='cells', reactants={'n': 1}, k=1.0)
>>> values = {'n': np.arange(3.0)}
>>> cells = FieldElement('cells', element=CompartmentElement, space=LatticeSpace((3,)), values=values)
>>> simulation = Simulation(System(elements=[cells], interactions=[decay]), dt=1.0).check().prepare()
>>> np.round(simulation.run(until=1).trajectory_.lattice('cells.n')[-1], 3).tolist()
[0.0, 0.368, 0.736]
A coefficient that is a parameter, the decay at the rate `x` that removes `1 / epsilon` units per event:
>>> from typing import Annotated
>>> from skcomplex.core.elements import Element
>>> from skcomplex.specs import ParameterSpec, StateSpec
>>> class ScaledElement(Element):
... def __init__(
... self,
... label,
... x: Annotated[float, StateSpec()] = 1.0,
... epsilon: Annotated[float, ParameterSpec()] = 0.5,
... ):
... super().__init__(label)
... self.x = x
... self.epsilon = epsilon
>>> scaled = ReactionInteraction(
... 'decay', target='box', reactants={'x': lambda s: 1 / s.target.epsilon}, propensity=lambda s: s.target.x
... )
>>> simulation = Simulation(System(elements=[ScaledElement('box')], interactions=[scaled]), dt=0.01)
>>> round(float(simulation.check().prepare().run(until=0.5).trajectory_.values('box.x')[-1]), 3)
0.368
Every agent of a group decays on its own states, by `k n` in a step of the rate equation:
>>> from skcomplex.core.elements import GroupElement
>>> from skcomplex.core.spaces import DiscreteTime
>>> class CellElement(Element):
... def __init__(self, label, n: Annotated[float, StateSpec()] = 3.0):
... super().__init__(label)
... self.n = n
>>> cells = GroupElement('cells', element=CellElement, size=2)
>>> decay = ReactionInteraction('decay', target='cells', reactants={'n': 1}, k=0.1, form='agent')
>>> system = System(elements=[cells], interactions=[decay], time=DiscreteTime())
>>> trajectory = Simulation(system).check().prepare().run(until=2).trajectory_
>>> np.round(trajectory.agents('cells', 2).values['n'], 6).tolist()
[2.43, 2.43]
"""
def __init__(
self,
label: str,
*,
target: Annotated[str | None, RoleSpec()] = None,
reactants: Annotated[Stoichiometry | None, RoleSpec(variadic=True)] = None,
products: Annotated[Stoichiometry | None, RoleSpec(variadic=True)] = None,
k: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
propensity: Callable[..., object] | None = None,
modifiers: Annotated[Stoichiometry | None, RoleSpec(variadic=True)] = None,
mode: str = 'rate',
form: str = 'symbolic',
phase: str | None = None,
schedule: str = 'synchronous',
) -> None:
super().__init__(label, target='')
self.target = target
self.reactants = reactants
self.products = products
self.k = k
self.propensity = propensity
self.modifiers = modifiers
self.mode = mode
self.form = form
self.phase = phase
self.schedule = schedule
def _get_reactants(self) -> Stoichiometry | None:
"""Return the coefficients of the reactants, which a subclass with roles of its own names returns."""
return self.reactants
def _get_products(self) -> Stoichiometry | None:
"""Return the coefficients of the products, which a subclass with roles of its own names returns."""
return self.products
def _get_modifiers(self) -> Stoichiometry | None:
"""Return the coefficients of the modifiers, which a subclass with roles of its own names returns."""
return self.modifiers
def _get_mode(self) -> str:
"""Return the form of the laws, which a subclass that sets it by another argument returns."""
return self.mode
def _get_bound(self, namespace: SimpleNamespace) -> dict[str, SimpleNamespace]:
"""Return the namespace of every path the plain roles of the reaction bind, by path."""
roles = [
name
for name, spec in self._get_specs().items()
if isinstance(spec, RoleSpec) and not spec.variadic and not spec.noise
]
return {
getattr(self, name): getattr(namespace, name)
for name in roles
if isinstance(getattr(self, name), str) and hasattr(namespace, name)
}
def _get_terms(
self,
entries: Stoichiometry | None,
namespace: SimpleNamespace,
name: str,
) -> list[tuple[Expression, Expression | float]]:
"""Return the state symbol and the coefficient of every entry of a kind of species.
A state name alone is a state of the target. A pair reads the namespace of the plain role that binds its path,
or else the namespace of its variadic role `name`, in the order of the pairs. A coefficient that is a function
is its expression at the namespace.
"""
entries = entries or {}
pairs = [entry for entry in entries if not isinstance(entry, str)]
sources: dict[object, SimpleNamespace] = {}
if pairs and name in self._get_specs():
sources = dict(zip(pairs, getattr(namespace, name), strict=True))
elif pairs:
bound = self._get_bound(namespace)
sources = {pair: bound[pair[0]] for pair in pairs}
return [
(
getattr(
self._get_member(namespace, 'target') if isinstance(entry, str) else sources[entry],
self._get_state(entry),
),
SR(coefficient(namespace)) if callable(coefficient) else coefficient,
)
for entry, coefficient in entries.items()
]
def _get_state(self, entry: tuple[str, str] | str) -> str:
"""Return the name of the state of an entry, a state name alone or the second item of a pair."""
return entry if isinstance(entry, str) else entry[1]
def _get_changes(self, namespace: SimpleNamespace) -> dict[Expression, float]:
"""Return the net change of every state the reaction changes, products minus reactants."""
changes: dict[Expression, float] = {}
for sign, entries, name in (
(-1, self._get_reactants(), 'reactants'),
(1, self._get_products(), 'products'),
):
for state, coefficient in self._get_terms(entries, namespace, name):
changes[state] = changes.get(state, 0) + sign * coefficient
return {state: change for state, change in changes.items() if change}
def _get_mass_action(self, namespace: SimpleNamespace, constant: Expression) -> Expression:
"""Return the mass action of a rate constant, times every reactant and modifier to its coefficient."""
propensity = SR(constant)
for entries, name in ((self._get_reactants(), 'reactants'), (self._get_modifiers(), 'modifiers')):
for state, coefficient in self._get_terms(entries, namespace, name):
propensity *= state**coefficient
return propensity
def _get_rate(self, namespace: SimpleNamespace) -> Expression:
"""Return the propensity of the reaction, by the user's function or by mass action at the rate constant `k`."""
if self.propensity is not None:
return SR(self.propensity(namespace))
return self._get_mass_action(namespace, self._get_parameter_symbol('k', namespace))
def _get_langevin_laws(self, propensity: Expression, changes: dict[Expression, float]) -> list[BasePart]:
"""Return the drift of the reaction and the noise of its one Wiener process on every state it changes."""
amplitude = sqrt(where(propensity > 0, propensity, 0))
rates: list[BasePart] = [RateLaw(state, change * propensity) for state, change in changes.items()]
noises: list[BasePart] = [
NoiseLaw(state, change * amplitude, process=self.label) for state, change in changes.items()
]
return rates + noises
def _is_entry(self, entry: object) -> bool:
"""Check whether an entry is a pair of a path and a state name, or a state name of a given target."""
if isinstance(entry, str):
return self.target is not None and entry.isidentifier()
return (
isinstance(entry, tuple)
and len(entry) == ENTRY_SIZE
and all(isinstance(item, str) for item in entry)
and entry[1].isidentifier()
)
def _is_coefficient(self, coefficient: object) -> bool:
"""Check whether a coefficient is a positive number or a function."""
return callable(coefficient) or (isinstance(coefficient, int | float) and coefficient > 0)
def _check_entries(self, name: str, entries: Stoichiometry | None) -> None:
"""Raise an error for entries that are neither pairs nor state names of the target, or not coefficients."""
if entries is None:
return
valid = isinstance(entries, Mapping) and all(
self._is_entry(entry) and self._is_coefficient(coefficient) for entry, coefficient in entries.items()
)
if not valid:
error_msg = (
f'The {name} of reaction {self.label!r} are {entries!r}. Map pairs of a path and a state name, or '
'state names of the target, to positive numbers or functions, such as {("S", "n"): 1} or {"n": 1}.'
)
raise ValidationError(error_msg)
def _check_form(self) -> None:
"""Raise an error for an unknown form, or an agent form without a target, with pairs, or of Langevin mode."""
if self.form not in FORMS:
error_msg = f'Reaction {self.label!r} has the form {self.form!r}. Use one of {", ".join(FORMS)}.'
raise ValidationError(error_msg)
self._check_variant('schedule', self.schedule, SCHEDULES)
if self.form != 'agent' and self.schedule != 'synchronous':
error_msg = f'The schedule {self.schedule!r} of reaction {self.label!r} runs agent rules. Use form="agent".'
raise ValidationError(error_msg)
if self.form != 'agent':
return
if self.target is None:
error_msg = f'Reaction {self.label!r} has the agent form but no target. Give the agent group as `target`.'
raise ValidationError(error_msg)
mode = self._get_mode()
if mode not in AGENT_MODES:
error_msg = (
f'Reaction {self.label!r} has the agent form and the mode {mode!r}. Use {" or ".join(AGENT_MODES)}.'
)
raise ValidationError(error_msg)
entries = [*(self._get_reactants() or {}), *(self._get_products() or {}), *(self._get_modifiers() or {})]
if not all(isinstance(entry, str) for entry in entries):
error_msg = (
f'Reaction {self.label!r} has the agent form, whose species are states of every member. Name them by '
'state names alone, such as {"n": 1}.'
)
raise ValidationError(error_msg)
def _check_jump(self) -> None:
"""Raise an error for an unknown mode or form, no reactants and products, or entries the reaction cannot take.
Raises:
ValidationError:
The mode or the form is unknown, the reaction has neither reactants nor products, an entry is neither a
pair of a path and a state name nor a state name of a given target, a coefficient is neither a positive
number nor a function, or the agent form has no target, has an entry that is a pair, or has the mode
`'langevin'`.
"""
mode = self._get_mode()
if mode not in MODES:
error_msg = f'Reaction {self.label!r} has the mode {mode!r}. Use one of {", ".join(MODES)}.'
raise ValidationError(error_msg)
reactants, products = self._get_reactants(), self._get_products()
if not reactants and not products:
error_msg = f'Reaction {self.label!r} has neither reactants nor products. Give at least one of them.'
raise ValidationError(error_msg)
for name, entries in (('reactants', reactants), ('products', products), ('modifiers', self._get_modifiers())):
self._check_entries(name, entries)
self._check_form()
def _get_number(self, value: object, agent: AgentViewProtocol, model: ModelViewProtocol) -> float:
"""Return a number, or the value of a function of an agent and the model as a number."""
found = value(agent, model) if callable(value) else value
if not isinstance(found, SupportsFloat):
error_msg = f'A function of reaction {self.label!r} returns {found!r} for an agent. Return a number.'
raise ValidationError(error_msg)
return float(found)
def _get_agent_changes(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> dict[str, float]:
"""Return the net change of every state of an agent the reaction changes, products minus reactants."""
changes: dict[str, float] = {}
for sign, entries in ((-1, self._get_reactants()), (1, self._get_products())):
for entry, coefficient in (entries or {}).items():
state = self._get_state(entry)
changes[state] = changes.get(state, 0.0) + sign * self._get_number(coefficient, agent, model)
return changes
def _get_agent_propensity(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> float:
"""Return the propensity of an agent, by the user's function or by mass action at the rate constant `k`."""
if self.propensity is not None:
return self._get_number(self.propensity, agent, model)
propensity = self._get_parameter_value('k', model)
for entries in (self._get_reactants(), self._get_modifiers()):
for entry, coefficient in (entries or {}).items():
value = float(getattr(agent, self._get_state(entry)))
propensity *= value ** self._get_number(coefficient, agent, model)
return propensity
def _get_agent_scale(self, propensity: float, model: ModelViewProtocol) -> float:
"""Return the multiple of its changes an agent adds in a step, its propensity or whether an event fires."""
if self._get_mode() == 'rate':
return propensity
return float(model.rng.random() < 1.0 - math.exp(-max(propensity, 0.0)))
def _get_agent_step(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> dict[str, float]:
"""Return the change of every state of an agent in the step, its changes times its propensity or its event."""
changes, propensity = self._get_agent_changes(agent, model), self._get_agent_propensity(agent, model)
scale = self._get_agent_scale(propensity, model)
return {state: change * scale for state, change in changes.items()}
def _apply_agent_step(self, agent: AgentViewProtocol, changes: dict[str, float]) -> None:
"""Add their changes to the states of an agent."""
for state, change in changes.items():
setattr(agent, state, float(getattr(agent, state)) + change)
def _react_member(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> None:
"""React in one agent on its own states, in its own activation."""
if agent.alive:
self._apply_agent_step(agent, self._get_agent_step(agent, model))
def react(self, model: ModelViewProtocol) -> None:
"""React in every agent of the target on its own states, by the schedule.
Args:
model:
The model view of the interaction.
"""
self._update_agents(model, self.schedule, self._get_agent_step, self._apply_agent_step)
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the laws of the reaction in its mode, or the rule of the agent form.
Args:
namespace:
The namespace of the symbols the laws can use, with the namespaces of the target, the reactants, the
products, and the modifiers.
Returns:
laws:
The rate laws, the jump law, the rate and noise laws, or the rule of the reaction.
"""
if self.form == 'agent' and self.schedule == 'member':
return [RuleLaw(self._react_member, phase=self.phase, members=self._get_role('target'))]
if self.form == 'agent':
return [RuleLaw(self.react, phase=self.phase)]
mode = self._get_mode()
if mode == 'jump':
return [*super()._get_laws(namespace)]
propensity = self._get_rate(namespace)
changes = self._get_changes(namespace)
if mode == 'langevin':
return self._get_langevin_laws(propensity, changes)
return [RateLaw(state, change * propensity) for state, change in changes.items()]
Equations
prostate.N: diff(N_prostate(t), t) == -(UT_prostate^2*phi1_n_prostate/(UT_prostate^2 + phi2_prostate^2) + UD_prostate^2*psi1_n_prostate/(UD_prostate^2 + psi2_prostate^2))*((c1_prostate*(AD_prostate(t) + AI_prostate(t)) + N_prostate(t))*Vc_prostate/((age_prostate + 1/8760*t)^phi_prostate*v1_prostate/((age_prostate + 1/8760*t)^phi_prostate + v2_prostate^phi_prostate) + v3_prostate) - 1)*N_prostate(t) - psi3_n_prostate*N_prostate(t)/(UD_prostate + UT_prostate + psi4_prostate)
prostate.AD: diff(AD_prostate(t), t) == (UT_prostate^2*phi1_d_prostate/(UT_prostate^2 + phi2_prostate^2) + UD_prostate^2*psi1_d_prostate/(UD_prostate^2 + psi2_prostate^2))*m1_prostate*((c1_prostate*N_prostate(t) + AD_prostate(t) + AI_prostate(t))/b2_prostate - 1)*AD_prostate(t) - (UT_prostate^2*phi1_d_prostate/(UT_prostate^2 + phi2_prostate^2) + UD_prostate^2*psi1_d_prostate/(UD_prostate^2 + psi2_prostate^2))*((c1_prostate*N_prostate(t) + AD_prostate(t) + AI_prostate(t))/b2_prostate - 1)*AD_prostate(t) - psi3_d_prostate*AD_prostate(t)/(UD_prostate + UT_prostate + psi4_prostate)
prostate.AI: diff(AI_prostate(t), t) == -(UT_prostate^2*phi1_d_prostate/(UT_prostate^2 + phi2_prostate^2) + UD_prostate^2*psi1_d_prostate/(UD_prostate^2 + psi2_prostate^2))*m1_prostate*((c1_prostate*N_prostate(t) + AD_prostate(t) + AI_prostate(t))/b2_prostate - 1)*AD_prostate(t) - (UT_prostate^2*phi1_i_prostate/(UT_prostate^2 + phi2_prostate^2) + UD_prostate^2*psi1_i_prostate/(UD_prostate^2 + psi2_prostate^2))*((c1_prostate*N_prostate(t) + AD_prostate(t) + AI_prostate(t))/b3_prostate - 1)*AI_prostate(t) - psi3_i_prostate*AI_prostate(t)/(UD_prostate + UT_prostate + psi4_prostate)
prostate.UIP: diff(UIP_prostate(t), t) == -gamma_p_prostate*(AD_prostate(t) + AI_prostate(t))^2*UIP_prostate(t)/(KP_prostate + AD_prostate(t) + AI_prostate(t)) - Vc_prostate*du1_prostate*N_prostate(t)*UIP_prostate(t)/((age_prostate + 1/8760*t)^phi_prostate*v1_prostate/((age_prostate + 1/8760*t)^phi_prostate + v2_prostate^phi_prostate) + v3_prostate) + (beta_d_prostate*AD_prostate(t) + beta_i_prostate*AI_prostate(t) + beta_n_prostate*N_prostate(t))*(UD_prostate + UT_prostate + sigma_prostate) - du2_prostate*UIP_prostate(t)
prostate.UP: diff(UP_prostate(t), t) == gamma_p_prostate*(AD_prostate(t) + AI_prostate(t))^2*UIP_prostate(t)/(KP_prostate + AD_prostate(t) + AI_prostate(t)) + Vc_prostate*du1_prostate*N_prostate(t)*UIP_prostate(t)/((age_prostate + 1/8760*t)^phi_prostate*v1_prostate/((age_prostate + 1/8760*t)^phi_prostate + v2_prostate^phi_prostate) + v3_prostate) - du3_prostate*UP_prostate(t)