This notebook shows how to study noise with MuMoT

There are 7 examples worked through below.

import mumot

mumot.__version__
mumot.about()

Production and decay of protein $P$ and its dimerization into $P_2$ ($P_2$ is called $Q$ below)¶

(see F. Hayot & C. Jayaprakash (2004), Physical Biology 1, pp. 205-210)

In [ ]:

model1 = mumot.parseModel(r"""
\emptyset -> P : k_3
P -> \emptyset : k
P + P -> Q + \emptyset : k_1
Q + \emptyset -> P + P : k_2
""")

In [ ]:

model1.showODEs(method='vanKampen')

In [ ]:

evol1 = model1.integrate(['Q', 'P'], legend_loc='center right')

In [ ]:

evol1.showLogs(tail=True)

In [ ]:

bookmark_evol1 = model1.integrate(showStateVars=['P', 'Q'], initialState = {'P': 1.0, 'Q': 0.0}, 
                                   maxTime = 3.0, plotProportions = False, 
                                   params = [('k_{3}', 2.0), ('k_{2}', 2.0), ('k', 2.0), ('k_{1}', 2.0), 
                                             ('plotLimits', 1), ('systemSize', 10.0)], 
                                   legend_loc = 'center right', bookmark = False)

In [ ]:

Ncorr1 = model1.noiseCorrelations(maxTime=120, legend_loc='upper right', legend_fontsize=10)

In [ ]:

Ncorr1.showLogs(tail=True)

In [ ]:

model1.noiseCorrelations()

In [ ]:

bookmark_ncorr1 = model1.noiseCorrelations(initialState = {'P': 1.0, 'Q': 0.0}, params = [('k_{3}', 2.0), ('k_{2}', 2.0), ('k', 2.0), ('k_{1}', 2.0), ('plotLimits', 1), ('systemSize', 10.0)], legend_loc = 'upper right', legend_fontsize = 10, maxTime = 120, bookmark = False)

In [ ]:

model1.showFokkerPlanckEquation()

In [ ]:

model1.showNoiseEquations()

In [ ]:

model1.showNoiseSolutions()

In [ ]:

stream1 = model1.stream('P', 'Q', showFixedPoints=True, showNoise = True)

In [ ]:

stream1.showLogs()

In [ ]:

bookmark_stream1 = model1.stream('P', 'Q', params = [('k_{3}', 2.0), ('k_{2}', 2.0), ('k', 2.0), ('k_{1}', 2.0), ('plotLimits', 1.5), ('systemSize', 10.0)], showFixedPoints = True, showNoise = True, bookmark = False)

The 'Brusselator' reaction mechanism (proposed by Prigogene and Lefever in 1968)¶

(see J.D. Murray (1991), Mathematical Biology, p. 175)

In [ ]:

model2 = mumot.parseModel(r"""
(\alpha) -> X : \gamma
X + X + Y -> X + X + X : \chi
(\beta) + X -> Y + \emptyset : \delta
X -> \emptyset : \xi
""")

In [ ]:

evol2 = model2.integrate(legend_loc='lower right')

In [ ]:

evol2.showLogs()

In [ ]:

evol2b = model2.integrate(['X'], maxTime=50, legend_loc='lower right')

In [ ]:

Ncorr2 = model2.noiseCorrelations(maxTime=20, legend_loc='upper right', legend_fontsize=14)

In [ ]:

Ncorr2.showLogs()

In [ ]:

model2.showFokkerPlanckEquation()

In [ ]:

model2.showNoiseEquations()

In [ ]:

model2.showNoiseSolutions()

In [ ]:

stream2 = model2.stream('X', 'Y', showNoise=True, showFixedPoints=False)

In [ ]:

stream2.showLogs()

Textbook example for multivariate master equation ¶

(see N. van Kampen (1981), Stochastic processes in physics and chemistry, pp. 273-277)

In [ ]:

model3 = mumot.parseModel(r"""
(A) -> X : \alpha
X + X -> Y + \emptyset : \gamma
Y -> (B) : \beta
""")

In [ ]:

model3.showODEs(method='vanKampen')

In [ ]:

evol3 = model3.integrate(legend_loc='center right')

In [ ]:

evol3.showLogs()

In [ ]:

Ncorr3 = model3.noiseCorrelations(maxTime=20, legend_loc='upper right', legend_fontsize=12)

In [ ]:

Ncorr3.showLogs()

In [ ]:

model3.showFokkerPlanckEquation()

In [ ]:

model3.showNoiseEquations()

In [ ]:

model3.showNoiseSolutions()

In [ ]:

stream3 = model3.stream('X', 'Y', showNoise=True, showFixedPoints=True)

In [ ]:

stream3.showLogs()

Pitchfork bifurcation: a prototypical example ¶

In [ ]:

model4 = mumot.parseModel(r"""
\emptyset + X -> X + X : \mu
X + X + X -> X + X + \emptyset : \alpha
Y -> \emptyset : \beta
(A) -> Y : \kappa
X -> \emptyset : \gamma
""")

In [ ]:

model4.showODEs(method='vanKampen')

In [ ]:

evol4 = model4.integrate(['X', 'Y'], legend_loc='center right')

In [ ]:

evol4.showLogs()

In [ ]:

Ncorr4 = model4.noiseCorrelations(maxTime=20, legend_loc='upper right', legend_fontsize=12)

In [ ]:

Ncorr4.showLogs()

In [ ]:

model4.showFokkerPlanckEquation()

In [ ]:

model4.showNoiseEquations()

In [ ]:

model4.showNoiseSolutions()

In [ ]:

stream4 = model4.stream('X', 'Y', showNoise=True, showFixedPoints=False)

In [ ]:

stream4.showLogs()

Time evolution of stop-signal model¶

In [ ]:

model5 = mumot.parseModel(r"""
U -> A : g_A
U -> B : g_B
A -> U : a_A
B -> U : a_B
A + U -> A + A : r_A
B + U -> B + B : r_B
A + B -> A + U : s
A + B -> B + U : s
""")

In [ ]:

evol5 = model5.integrate(['A', 'B', 'U'], maxTime=50, legend_loc='center right', legend_fontsize=13)

In [ ]:

evol5.showLogs()

In [ ]:

Ncorr5 = model5.noiseCorrelations(maxTime=50, legend_loc='upper right', legend_fontsize=12)

In [ ]:

Ncorr5.showLogs()

In [ ]:

model5.showNoiseEquations()

Stop-signal model with 3 options¶

In [ ]:

model6 = mumot.parseModel(r"""
U -> A : g_1
U -> B : g_2
U -> C : g_3
A -> U : a_1
B -> U : a_2
C -> U : a_3
A + U -> A + A : r_1
B + U -> B + B : r_2
C + U -> C + C : r_3
A + B -> A + U : s
A + B -> B + U : s
A + C -> A + U : s
A + C -> C + U : s
B + C -> B + U : s
B + C -> C + U : s
""")

In [ ]:

evol6 = model6.integrate(['A', 'B', 'C'], maxTime=50, legend_loc='center right')

In [ ]:

evol6.showLogs()

Population control of E. coli cells¶

(see You et al. (2004), Nature 428, pp. 868-871)

In [ ]:

model7 = mumot.parseModel(r"""
A -> E : k_E
\emptyset + X -> X + X : k
\emptyset + X -> X + X : v_A
X + X -> X + \emptyset : k_m
E -> \emptyset : d_E
X -> A : v_A
A -> \emptyset : d_A
E + X -> E + \emptyset : d_N
A + \emptyset -> A + A : k_E
""")

In [ ]:

model7.showODEs(method='vanKampen')

In [ ]:

evol7 = model7.integrate(['A', 'E', 'X'], maxTime=50, legend_loc='center right')

In [ ]:

evol7.showLogs()

In [ ]:

Ncorr7 = model7.noiseCorrelations(maxTimeDS=100, tstep=0.02, maxTime=20, 
                                  legend_loc='upper right', legend_fontsize=16)

In [ ]:

Ncorr7.showLogs()

In [ ]:

model7.showFokkerPlanckEquation()

In [ ]:

model7.showNoiseEquations()

This notebook shows how to study noise with MuMoT

Production and decay of protein $P$ and its dimerization into $P_2$ ($P_2$ is called $Q$ below)¶

(see F. Hayot & C. Jayaprakash (2004), Physical Biology 1, pp. 205-210)

The 'Brusselator' reaction mechanism (proposed by Prigogene and Lefever in 1968)¶

(see J.D. Murray (1991), Mathematical Biology, p. 175)

Textbook example for multivariate master equation¶

(see N. van Kampen (1981), Stochastic processes in physics and chemistry, pp. 273-277)

Pitchfork bifurcation: a prototypical example¶

Time evolution of stop-signal model¶

Stop-signal model with 3 options¶

Population control of E. coli cells¶

(see You et al. (2004), Nature 428, pp. 868-871)

Textbook example for multivariate master equation ¶

Pitchfork bifurcation: a prototypical example ¶