To look for conditions for bistability, a simplified system is studied where ScbA protein
can bind to its operator OA' and activate its own transcription. This obviates the
requirement of formation of ScbA-ScbR complex that activates scbA. In addition, the
growth rate is embedded in the decay rates of all species, and the basal rate of production
of scbA is neglected. The degradation of ScbR protein is also neglected. The resulting set
of equations is listed below:
KOR
d [r ]
 kmR
 kdr [r ]
dt
KOR  [ R]
(1.1)
KOA  [ A]
d [a]
 kmA
 kda .[a]
dt
( KOA  [ R])( KOA'  [ A])
(1.2)
d [ R]
 k pR [r ]  kbCR [Ci ][ R]  kbCR [CR]
dt
(1.3)
d [ A]
 k pA .[a ]  kdA .[ A]
dt
(1.4)
d [CR ]
 kbCR [Ci ][ R]  (kbCR  kdCR )[CR ]
dt
(1.5)
d [Ci ]
 kC .[ A]  (kbCR .[Ci ].[ R]  kbCR .[CR])  k se .([Ci ]  [Ce ])
dt
(1.6)
To determine the steady state solutions, the L.H.S of the above equations is set to zero.
kmR
K OR
kdr K OR  [ R ]
k
KOA  [ A]
[a]  mA
kda ( KOA  [ R])( KOA'  [ A])
[r ] 
[CR]  CR [Ci ][ R]
R
KOR  [ R]
(1.7)
(1.8)
kdCR kbCR
(kbCR  kdCR )
k pR kmR KOR
where  R 
kdr
where  CR 
  CR [Ci ][ R] ,
 A  [ A]
( KOA  [ R])( KOA'  [ A])
 kdA .[ A] where  A 
kC .[ A]  CR [Ci ][ R]  kse .([Ci ]  [Ce ]) .
k pA .kmA .KOA
kda
(1.9)
(1.10)
(1.11)
(1.12)
1
Equation (1.11) suggests that one steady state of the system is when [ A]  0 . The
kse [Ce ]
corresponding value for [Ci ] is
. [R] is given by the roots of the following
 CR [ R]  kse
cubic equation:
kdR CR [ R]3  (kdR K OR CR  kdR k se   CR k se [Ce ])[ R]2  ( CR k se [Ce ]K OR  kdR K OR k se   CR k pR K OR )[ R ]
kse k pR K OR  0
(1.13)
Physiologically, the steady state with [A] = 0 corresponds to the OFF state where a high
concentration of ScbR protein represses transcription of scbA. This results in low levels
of CR complex. The dominant characteristic of the system in this state is that the autorepressed synthesis of ScbR is balanced by the low levels of CR complex formation. The
stability condition for this steady state is:
kda kdA 
kmA k pA  KOA 

0
KOA '  KOA  [ R] 
(1.14)
For a fixed set of parameter values, the coefficients of equation 1.13 are a function of
[Ce]. [R] is observed to decrease with increasing Ce. At low Ce, [R] is high and the above
stability
condition
if [ R] 
A
kdA KOA '
is
satisfied.
However,
this
steady
state
is
unstable
 KOA .
The other steady states corresponding to [ A]  0 can be determined by solving the
following equations based on 1.10-1.12 above:
A
kdA ( KOA  [ R])
 ( KOA'  [ A])
(1.15)
kC .[ A]  CR [Ci ][ R]  kse .[Ci ]  kse .[Ce ]
(1.16)
 R  CR [Ci ][ R]( KOR  [ R])
(1.17)
The above equations simplify to:
2
(kdA K OA' 
kse
kdA [Ce ])[ R]3
kC
(kdA K OA' K OR   A  kdA K OA' K OA 
(
kdA R kse

kdA [Ce ]( K OR  K OA ))[ R ]2
kC
kC
kse
k 
k  k K
kdA [Ce ]K OA K OR   A K OR  kdA K OR K OA' K OA  dA R ( CR K OA  k se ))[ R ]  dA R se OA  0
kC
 CR kC
 CR kC
(1.18)
The
product
of
three
roots
 R kse KOA kse
/ ( [Ce ]  K OA )
 CR kC
kC
'
(( KOA' 
for
the
and
the
above
cubic
sum
of
equation
roots
is
is
given
given
by
by
kse
k

[Ce ])( KOR  K OA )   A  R ) / ( se [Ce ]  K OA' ) . The above equation will have
kC
kC
kC
two positive roots (and one negative root) of [R] if the product is negative and the sum of
roots
is
( KOA' 
positive
i.e.
if
K OA' 
kse [Ce ]
0
kC
(or
[Ce ] 
K OA' kC
kse
)
and
kse

[Ce ])( KOR  KOA )  R   A .
kC
kC
For [Ce ] 
KOA' kC
kse
the product of roots will be positive and the sum will be negative i.e.
there is one positive root of [R]. For physiologically feasible steady states, concentration
of all other species should also be positive. From equations 1.15 and 1.16
[ A] 
A
kdA ( K OA  [ R])
 K OA' and [Ci ] 
kC .[ A]  kse .[Ce ]
. Therefore, while [Ci ] is always
 CR [ R]  kse
positive, [A] will be positive only for [ R] 
A
kdA KOA'
 KOA . Note, that this is also the
stability condition for the fixed state corresponding to [A]=0.
3
Thus, for [ A]  0 the system will have two positive fixed points if [Ce ] 
[ R] 
A
kdA KOA'
 KOA . Beyond [Ce] values where [ R] 
A
kdA KOA'
K OA' kC
kse
and
 KOA there is only one
fixed state.
An analytical condition for the stability of the roots is difficult to determine. A stable
steady state requires all the eigen values of the Jacobian matrix to be negative. The trace
of the Jacobian matrix for equation 1.18 is always negative. The determinant will be
negative if the following condition is satisfied:
2
kda kdA
( KOA  [ R])(kse   [ R])(
  kmA KOA k pA [ R](
R
( KOR  [ R])
2

 (kse [Ce ]  kC [ A])
)
kse   [ R]
kC kdA
 k k 2 ( K  [ R])(kse [Ce ]  kC [ A])
 da dA OA
)
( KOA  [ R])
kmA KOA k pA (kse   [ R])
Negative trace and determinant is a necessary but not sufficient condition for all eigen
values to be negative.
To summarize, the simplified ScbA-ScbR system represented by equations 1.1 - 1.6 will
have three positive fixed points for increasing [Ce] until the transition point that satisfies
[ R] 
A
kdA KOA'
 KOA . One fixed point corresponds to [A]=0, whereas the other two are
roots of equation 1.18 where [ A]  0 . For [Ce] values where [ R] 
A
kdA KOA'
 KOA , there
are two fixed points only one of which can be stable.
Figure S5A illustrates the steady states for a set of parameter values that satisfy the above
conditions. It was observed that one of the two roots corresponding to lower value of [R]
was stable and the other unstable. This stable steady state corresponding to a low [R]
results in low degree of suppression of scbA. This in turn leads to high values of A, Ci, Ce
and CR complex, which represents the ON state for expression of cpk cluster genes. In
this case, the production of ScbR is balanced by the formation of SCB1-ScbR (CR)
complex.
4
Note that the basal transcription rate of scbA was neglected in this analysis. If the basal
rate is accounted for, there is only one change in the steady state plot. The unstable steady
state corresponding to [A] = 0 disappears beyond the transition point. The rest of the
steady states are not affected. The steady state diagram is shown in Figure S5B.
5