Supplementary information Derivation, identification and validation

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Supplementary information
Derivation, identification and validation of a computational model of a
novel synthetic regulatory network in yeast.
Marucci L, Santini S, di Bernardo M, di Bernardo
D
Derivation of the DDAEs model
Model B in the main text contains two not measurable quantities, [Gal4Gal80] and [GALGal80].
A way to simplify the modelling of the mechanisms induced by the galactose is to make a steadystate approximation for such quantities. This approximation is reasonable, since the formation
of protein complexes occurs at a faster time scale with respect to transcription. By setting to
0 the left-hand side of equations (13) and (14) in the main text, and taking into account the
assumption of steady state for the protein levels of the network genes ([A9] in the main text),
we obtain:
[Gal4Gal80] =
K1
[Gal4f ree ][Gal80f ree ],
K2
(1)
K3
[GAL][Gal80f ree ].
K4
(2)
[GALGal80] =
Substituting (1) in eq. (11) of the main text we have:
[GAL4 ] = [Gal4free ] +
K1
[Gal4free ][Gal80free ],
K2
and thus
L. Marucci
Telethon Institute of Genetics and Medicine (TIGEM), Naples 80131, Italy .
Tel: +39 081 6132319; Fax: +39 081 6132 351
Department of Computer and Systems Engineering, Federico II University, Naples 80125, Italy.
Tel: +39 081 7683909; Fax: +39 081 7683186
S. Santini
Department of Computer and Systems Engineering, Federico II University, Naples 80125, Italy
M. d. Bernardo
Department of Computer and Systems Engineering, Federico II University, Naples 80125, Italy,
E-mail: [email protected]
D. di Bernardo
Telethon Institute of Genetics and Medicine (TIGEM), Naples 80131, Italy
Department of Computer and Systems Engineering, Federico II University, Naples 80125,
E-mail: dibernardo.tigem.it
(3)
2
[Gal4f ree ] =
[GAL4 ]
1+
f ree
K1
]
K2 [Gal80
.
(4)
The concentration of Gal80f ree free can now be written as a function of the parameters involved
in the formation of the complexes and of the concentrations of galactose, the free amount of
Gal4, the total amount of GAL80 and total amount of GAL4 :
[Gal80f ree ] = f ([GAL], [Gal4f ree ], [GAL80 ], [GAL4 ], K1 , K2 , K3 , K4 ).
(5)
Substituting (4) into (1), we obtain:
[Gal4Gal80] =
K1
[GAL4 ]
[Gal80f ree ],
f ree
1
K2 1 + K
[Gal80
]
K2
and then, substituting (6) and eq. (2) in eq. (12) of the main text, we have:
!
K3
[GAL4 ]
free
[GAL80 ] = [Gal80 ] 1 +
[GAL] + K
.
free
2
K4
]
K1 + [Gal80
(6)
(7)
This leads to the following constraint:
K2
K3
K2 K3
f ree 2
f ree
[Gal80
] 1+
[GAL] + [Gal80
]
+ [GAL4 ] − [GAL80 ] +
[GAL]
K4
K1
K1 K4
K2
−
[GAL80 ] = 0 . (8)
K1
Thus, considering the modelling assumptions [A9] and [A10] described in the main text,
the DDAEs model consists of the following five DDEs plus the algebraic non linear constraint in
equation (8).
!
!
k2h2
[SWI5 (t − τ )]h1
·
− d1 [CBF1 ],
k1h1 + [SW I5(t − τ )]h1
k2h2 + [ASH1 ]h2
!
d[GAL4 ]
[CBF1 ]h3
= α2 + v2
− (d2 − ∆(ψ1 ))[GAL4 ],
dt
k3h3 + [CBF1 ]h3
 
h4
d[CBF1 ]
= α1 + v1
dt

d[SWI5 ]
= α3 + v3 

dt
[GAL4 ]
K
1+ K1 [Gal80f ree ]
2
k4h4 +
[GAL4 ]
[Gal80f ree ]

h 4 
 − d3 [SWI5 ],
(9)
(10)
(11)
K
1+ K1
2
d[GAL80 ]
= α4 + v4
dt
[SWI5 ]h5
h5
k5 + [SWI5 ]h5
!
d[ASH1 ]
= α5 + v5
dt
[SWI5 ]h6
k6h6 + [SWI5 ]h6
!
− (d4 − ∆(ψ2 ))[GAL80 ],
(12)
− d5 [ASH1 ].
(13)
3
Supplementary Figure 1 (A) Schematic representation of galactose pathway. When the galactose medium is
present, the activated Gal3 alters the free concentration of Gal80 through sequestration in the cytoplasm, thus
relieving its inhibition on Gal4. In presence of glucose, the dimerized form of Gal80 directly to the Gal4 dimer.
(B) Simplified representation of the galactose induced switch described by models A and B. Here it’s assumed
that Gal80 directly binds to galactose when the network is on while Gal4 and Gal80 form the complex Gal4Gal80
without any prior dimerization when the network is off.
4
Supplementary Figure 2 Identification results on time-series data. Circles represent average expression data
for each of the IRMA genes at different time points. Dashed lines represent standard errors. Continuous colored
lines represent in silico data. (A) Identification results of the model B2 on the preliminary ”Glucose-to-Galactose”
time-series. (B) Identification results of the model B3 on the preliminary ”Glucose-to-Galactose” time-series. (C)
Identification results of the model B4 on the preliminary ”Glucose-to-Galactose” data-set.
5
Supplementary Figure 3 (A), (B) In silico expression levels of IRMA genes obtained by simulating the overexpression of each gene with the model B3. The black dots on the bars indicate the perturbed gene. White and
grey bars represent steady states of the genes respectively in unperturbed and perturbed conditions.
6
Supplementary Figure 4 Identification results on time-series data. Circles represent average expression data
for each of the IRMA genes at different time points. Dashed lines represent standard errors. Continuous colored
lines represent in silico data. (A) Identification results of the model D on the ”Glucose-to-Galactose” time-series
(average of 5 time-series). (B) Identification results of the model D on the ”Glucose-to-Galactose” time-series.
(C) Validation results of the model D on the ”Glucose-to-Galactose” data-set (average of 4 time-series).