Fixpoints¶

Fixpoints are global states of the networks where no transition is possible. Pint implements the complete fixpoints computation in automata networks as Boolean satisfaction constraints (SAT). It should be tractable on networks with hundreds of components. Usually, the main limitation is the potential very high number of fixpoints.

Let us illustrate the fixpoint computation on a medium size example:

[2]:

erbb = pypint.load("models/ERBB_G1-S.an")
len(erbb.automata)

[2]:

20

The method fixpoints() returns the list of all the fixpoints of the network. The result is independent from the initial state. In the following example, we use Pandas to provide a pretty display of the returned list. This is purely cosmetic.

[3]:

fps = erbb.fixpoints()
import pandas as pd # for pretty display of fixpoints
pd.DataFrame(fps)

[3]:

Returned fixpoints can be set as initial state of the network as follows:

[4]:

erbb_fp = erbb.having(fps[1]) # copy the model with the second fixpoint of the list as initial state
erbb_fp.initial_state.nonzeros()

[4]:

{'AKT1': 1,
 'CDK2': 1,
 'CDK4': 1,
 'CDK6': 1,
 'CycD1': 1,
 'CycE1': 1,
 'ERalpha': 1,
 'IGF1R': 1,
 'MEK1': 1,
 'MYC': 1,
 'pRB': 1}

Reachable state graph¶

Pint also implements explicit reachable state graph analysis from the model initial state. Note that such a kind of approach is limited in scalability. On classical desktop computers, the state graph analysis can handle mostly up to a few (reachable) million states.

The method count_reachable_states() returns the number of states that can be reached from the initial state. The implementation only relies on sets of reachable states and do not store transitions, which increases the tractability.

[5]:

erbb.initial_state.nonzeros()

[5]:

{'EGF': 1}

[6]:

erbb.count_reachable_states()

[6]:

2963

The method reachable_attractors() returns the complete list of attractors reachable from the initial state. An attractor is described by a type (fixpoint or cyclic), a size, and an instance of state belonging to the attractor.

[7]:

erbb.reachable_attractors()

[7]:

[{'sample': {'AKT1': 1,
   'CDK2': 1,
   'CDK4': 1,
   'CDK6': 1,
   'CycD1': 1,
   'CycE1': 1,
   'EGF': 1,
   'ERBB1': 1,
   'ERBB1_2': 1,
   'ERBB1_3': 1,
   'ERBB2': 1,
   'ERBB2_3': 1,
   'ERBB3': 1,
   'ERalpha': 1,
   'IGF1R': 0,
   'MEK1': 1,
   'MYC': 1,
   'p21': 0,
   'p27': 0,
   'pRB': 1},
  'size': 1,
  'type': 'fixpoint'}]

[8]:

phage = pypint.load("models/phageLambda.an")
attractors = phage.reachable_attractors()
attractors

[8]:

[{'sample': {'CI': 2, 'CII': 0, 'Cro': 0, 'N': 0},
  'size': 1,
  'type': 'fixpoint'},
 {'sample': {'CI': 0, 'CII': 0, 'Cro': 2, 'N': 0},
  'size': 2,
  'type': 'cyclic'}]

Finally, the method reachable_stategraph() returns a NetworkX digraph of the reachable state transition graph. Note that the performance can be tempered by the data exchange between the Pint binary and the python module.

If the graph is of small size, it can be displayed using GraphViz. The initial state has a gray background.

[9]:

metazoan = pypint.load("models/metazoan.an")
metazoan.reachable_stategraph()

[9]:

[10]:

phage.having(attractors[1]["sample"]).reachable_stategraph() # display the cyclic attractor of the Lambda phage model

[10]:

Related functions and classes¶

Model.fixpoints(timeout=None)

Returns the complete list of fixed points of the model.

Parameters

timeout (int) – command timeout in seconds

Return type

dict list

Model.count_reachable_states(tool=’pint’, timeout=None)

Counts the exact number of states reachable from initial_state. Uses an explicit state space approach.

Parameters

• tool (str) –

  • "pint" explicit reachable state space computation (default);

  • "its" symbolic reable stae space computation

• timeout (int) – command timeout in seconds

Return type

int

Model.reachable_stategraph(timeout=None)

Returns the reachable state graph from initial_state.

Parameters

timeout (int) – command timeout in seconds

Return type

NetworkX digraph (nx.DiGraph)

Model.reachable_attractors(timeout=None)

Returns the complete list of attractors reachable from initial_state.

Uses an explicit state space exploration methods.

Parameters

timeout (int) – command timeout in seconds

Each attractor is described by a dict object with the following keys:

• "type": either "fixpoint" or "cyclic".

• "size": number of states in the attractor (1 if fixpoint, >1 if cyclic).

• "sample": state (represented as dict) belonging to the attractor, i.e., either the fixpoint, or one of the state in the cycle attractor.