www.sciencemag.org/cgi/content/full/325/5940/542/DC1
Supporting Online Material for
Mathematical Biology Education: Beyond Calculus
Raina Robeva* and Reinhard Laubenbacher
*To whom correspondence should be addressed. E-mail: [email protected]
Published 31 July 2009, Science 325, 542 (2009)
DOI: 10.1126/science.1176016
This PDF file includes
SOM Text
Figs. S1 and S2
References
MATHEMATICAL BIOLOGY EDUCATION: BEYOND CALCULUS
RAINA ROBEVA AND REINHARD LAUBENBACHER
Abstract. This document contains supporting online text for the article with the
same title in the journal Science. Here, we provide some additional information and
mathematical detail relevant to the mathematical models of the lac operon discussed
in the article, their dynamic behavior, and their abilities to capture two key properties
of the lac operon system regulation: the system being ON and OFF and the feature
of bistability.
Key Features of the Lac Operon System
The lactose (lac) operon is a gene regulatory mechanism that controls the transport
and metabolism of lactose in E. coli. Since the seminal work by Jacob and Monod
in 1961 [S1], the lac operon has become one of the most widely studied and best understood mechanisms of gene regulation. It has also been used as a test system for
virtually any mathematical method for modeling of gene regulation, including differential equations (DE) models, algebraic models, stochastic models, and simulations.
When glucose is present in the cell and lactose is absent, the lac-repressor binds
to the operator region of the operon, and RNA polymerase, bound to the promoter,
is unable to move past this region. Hence, no transcription of the lac genes occurs.
In this case, the operon is OFF. In the absence of glucose, extracellular lactose is
transported into the cell by permease. Once inside the cell, lactose is converted into
glucose, galactose, and allolactose by the action of β-galactosidase. Allolactose is the
inducer of the lac operon, binding to the lac-repressor and inducing a conformational
change that prevents the repressor from binding to the operator region. The RNA
polymerase is able to move along the DNA, transcription of the lac genes occurs, and
lactose is metabolized. In this case the operon is ON.
Bistability is a system’s ability to achieve two different steady states based on the
same set of external outputs [S5]. In the case of the lac operon, it has been known since
the fifties [S4] that it exhibits a hysteresis effect: In the absence of glucose, the operon
is uninduced for low concentrations below a threshold L1 of extracellular lactose and
fully induced at high concentrations above a threshold of L2 . In the interval [L1 , L2]
both induced and uninduced cells can be observed and their status depends on the cell
history (the system response is hysteresis). The interval [L1, L2 ] is defined as a region
of bistability. Cells grown in an environment poor in extracellular lactose will have low
levels of internal lactose and allolactose and may remain uninduced for lactose levels in
the interval [L1 , L2 ]. In contrast, cells grown in a lactose-rich environment may remain
1
2
RAINA ROBEVA AND REINHARD LAUBENBACHER
induced for concentrations in the interval [L1 , L2 ]. A recent discussion of the bistability
feature of the lac operon together with green-fluorescence and inverted phase-contrast
images of a cell population showing a bimodal distribution of lac expression levels can
be found in [S5]. A survey of the quantitative approaches to the study of bistability in
the lac operon is given in [S7].
The modeling process begins with identifying the major interactions and components
of the system depicted in Figure S1 [S7], choosing the model variables and parameters,
and constructing a wiring diagram in the form of a directed graph (article figure, left
panel). Each compartment in the shaded part of this diagram is a variable in the model
and the compartments outside of the shaded region are parameters. The directed links
represent the influences between the variables. A positive influence is indicated by an
arrow, while a negative influence is depicted by a circle.
We now describe in more detail the mathematical models depicted in the article
figure (middle and right panels).
A DE Model of the Lac Operon
To construct a DE model, one needs to have detailed knowledge of the interactions
between the model variables. These may include the specifics of the control mechanisms, rates of production and degradation, highest and lowest biologically relevant
concentrations, and delays in the interactions between the components. These values
are included in the form of parameters. When, as is often the case, detailed information
is unavailable for all interactions of the system, some assumptions need to be made.
For instance, control mechanisms can generally be represented by any monotone function that takes values in the interval [0, 1] and includes parameters corresponding to
the potency of its influence. Hill functions are often chosen to represent control but, in
principle, an analysis of the model should be included in case Hill functions have been
chosen just as one of many alternatives. In the process of model validation, unknown
parameters are estimated by fitting the model to experimental data. This step should
generally be followed by a careful goodness-of-fit evaluation and sensitivity analysis.
The DE model depicted in the article figure (middle panel) is a minimal DE model
of the lac operon involving three variables: the intracellular concentration of mRNA
(M), lacZ polypeptide (E), and intracellular lactose (L). Since β-galactosidase is a
homo-tetramer made up of four identical lacZ polypeptides and the translation rate of
the lacY transcript is assumed to be the same as the rate for the lacZ transcript, the
following holds for the intracellular concentrations of β-galactocidase (B) and pemiase
(Q): Q = E and B = E/4. The model also assumes that the concentrations of internal
lactose (L) and allolactose (A) are the same. This model is justified and analyzed in
[S6].
MATHEMATICAL BIOLOGY EDUCATION: BEYOND CALCULUS
3
The full set of equations for the model of the lac operon are:
Ṁ = DkM PD (Ge )PR (A) − γM M
kE M − γE E
kL βL (Le )βG (Ge )Q − 2φM M(L)B − γL L
L
E
E/4
pp (1 + pc (Ge )(kpc − 1))
PD (Ge ) =
1 + pp pc (Ge )(kpc − 1)
KGnh
pc (Ge ) =
KGnh + Gne h
1
PR(A) =
ξ123 ρ(A)
1 + ρ(A) + (1+ξ2 ρ(A))(1+ξ
3 ρ(A))
KA 4
ρ(A) = ρmax
KA + A
Le
βL(Le ) =
kL + Le
Ge
βG (Ge ) = 1 − φG
kG + Ge
L
M(L) =
kM + L
Ė
L̇
A
Q
B
=
=
=
=
=
The differential equations in the model govern the time evolution of the variables
M, E, and L. This DE model also contains a large number of parameters, among
which Ge and Le stand for the concentrations of external glucose and lactose. Detailed
descriptions for the rest of the model parameters and functions can be found in [S6],
where the authors show that this model can exhibit bistability.
A Boolean Model of the Lac Operon
The Boolean model depicted in the article figure (right panel) is a direct translation
of the wiring diagram and of the biological interactions it represents into the set of
equations below. When two or more variables influence a third variable simultaneously,
we use the logical construct AND (denoted by the ∧ symbol ). To reflect that either of
two variables can exert its effect independently, we use OR (denoted by the ∨ symbol).
NOT (denoted by the ¬ symbol ) indicates a negative effect.
4
RAINA ROBEVA AND REINHARD LAUBENBACHER
fM = ¬Ge ∧ (L ∨ Le )
fE = M
fL = ¬Ge ∧ E ∧ Le
Time is discretized and at each time-step t the system is represented by the binary 3tuple (Mt , Et , Lt ), where each component stands for the value of the respective Boolean
variable at time t. The system update from time t to time t + 1 is done according to
the functions f on the left that represent the regulation of the individual variables.
For instance, the equation fL = ¬Ge ∧ E ∧ Le indicates that internal lactose L will be
present at time t + 1 (that is, L = 1) if Ge is not present at time t and E and Le are
both present at time t. Updating is performed synchronously on the three variables for
any given state of the system and the subsequent transitions represent the trajectory
of this state.
Given the small number of variables, we can examine the full state space of the
system for all possible choices of the parameters: (Le , Ge ) = (0, 0), (0, 1), (1, 0) and
(1, 1) (Figure S2 below). Figure S2 shows that for any choice of the parameters Le
and Ge , the system has a unique steady state and no limit cycles. The states (1, 1, 1)
and (0, 0, 0) correspond to the operon being ON and OFF, respectively. Figure S2
demonstrates that the operon is ON only when external lactose is available (Le = 1)
and external glucose is not (Ge = 0).
The Boolean model under consideration arises as a reduction of a more complex
model of the lac operon network discussed in [S3] and [S8], containing nine variables
and depending upon the parameters Le and Ge . In [S8] the authors show that this more
complex model and its reduced versions exhibit bistability after introducing stochasticity in the uptake of lactose for a group of cells. In particular, the following minimal
model that can be obtained by eliminating the variable E from the Boolean model
above exhibits bistability [S8]:
fM = ¬Ge ∧ (L ∨ Le )
fL = ¬Ge ∧ M ∧ Le
MATHEMATICAL BIOLOGY EDUCATION: BEYOND CALCULUS
5
References
[S1]
[S2]
[S3]
[S4]
[S5]
[S6]
[S7]
[S8]
F. Jacob, J. Monod. Genetic regulatory mechanisms in the synthesis of proteins. J. Mol.
Biol. v. 3, 318-356 (1961).
A.
Jarrah,
et
al.
DVD:
Discrete
Visualizer
of
Dynamics.
http://dvd.vbi.vt.edu/network visualizer/.
R. Laubenbacher, B. Sturmfels. Computer Algebra in Systems Biology. arXiv:0712.4248v2
(2008). To appear in American Mathematical Monthly.
A. Novick, M. Weiner. Enzyme induction as an all-or-none phenomenon. Proc. Natl. Acad.
Sci. U.S.A. v. 43, n. 7, 553-566 (1957).
E. Ozbudak et al. Multistability in the lactose utilization network of Escherichia coli. Nature
v.427, 737-340 (2004).
M. Santillan, et al. Origin of bistability in the lac operon. Biophys. J. v. 92, 11, 3830-3842
(2007). http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1868986.
M. Santillan, M. Mackey. Quantitative approaches to the study of bistability in the lac
operon of Escherichia coli. J. R. Soc. Interface v. 5, S29-S39 (2008).
B. Stigler, A. Veliz-Cuba. Network Topology as a Driver of Bistability in the Lac Operon.
arXiv:0807.3995 (2008).
Department of Mathematical Sciences, Sweet Briar College, VA 24595
E-mail address: [email protected]
Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA 24061
E-mail address: [email protected]
Fig. S1. Schematic of the lac operon regulatory mechanisms. This operon consists of genes lacZ,
lacY and lacA. Protein LacY is a permease that transports external lactose into the cell. Protein
LacZ polymerizes into a homotetramer named beta-galactosidase. This enzyme transforms
internal lactose (Lac) to allolactose (Allo) or to glucose and galactose (Gal). It also converts
allolactose to glucose and galactose. Allolactose can bind to the repressor R inhibiting it. When
not bound by allolactose, R can bind to a specific site upstream of the operon structural genes
and thus avoid transcription initiation. External glucose inhibits the production of cAMP that,
when bound to protein CRP to form complex CAP, acts as an activator of the lac operon.
External glucose also inhibits lactose uptake by permease proteins.
Figure reprinted from Biophys. J. 92, 11, M. Santillán, M.C. Mackey and E.S. Zeron, Origin of
Bistability in the lac Operon, 3830-3842 (2007), with permission from Elsevier.
Le = 0 and Ge = 0. The operon is OFF.
Le = 0 and Ge = 1. The operon is OFF.
Le = 1and Ge = 0. The operon is ON
Le = 1 and Ge = 1. The operon is OFF.
Fig. S2. The state space diagrams of the Boolean model for the four possible values of the
parameters Le and Ge. The diagrams are created using the DVD software [S2].