Download Generalized Circuit Analysis of Genetic Networks

Generalized Circuit Analysis of Biological Networks
Momiao Xiong1*, Jonathan Arnold2
1
Human Genetic Center, University of Texas Health Science Center at Houston, Houston,
TX 77030
Department of Genetics, University of Georgia, Athens, GA 30602
*Address for correspondence and reprints: Dr. Momiao Xiong, Human Genetics Center, University of
Texas - Houston, P.O. Box 20334, Houston, Texas 77225, (Phone): 713-500-9894, (Fax): 713-500-0900,
E-mail: [email protected]
Abstract
Comprehensive knowledge about a living system requires an understanding of the underlying
chemical reaction network of the cell. Using biological networks to model biological systems is
not a new approach1, but biological networks have many unique features and pose a significant
challenge because of their scale and unprecedented complexity. It is therefore important to have a
novel conceptual framework to describe quantitatively a network’s properties as well as new
methods for integrating experimental data and theories in biological networks. To accomplish
these tasks, we develop a linear circuit theory as a general framework for genetic network
analysis. We first introduce concepts of transcriptional potential, current, and resistance that can
be used to quantify the operation of a genetic network. Kirchhoff’s Current Law and
Ohm’s Law are fundamental to electrical circuits. Applications of linear circuit theory to
genetic networks lead to similar laws in a genetic network. Technologies for measuring
resistance of a transcript have not been developed. To overcome this, we will develop
algorithms to estimate transcriptional resistance based on a structural equation model for
genetic networks. Finally, to illustrate the generalized circuit analysis of genetic networks,
the proposed models and algorithms are applied to a part of an apoptosis network in
humans.
Introduction
A cell is a complex biological system consisting of many interrelated components.
Molecular biology in the last century has mainly focused on taking apart individual cellular
components, until we come face to face with the entire genetic blueprint or the entire proteome of
an organism. Although genes are of primary importance, the function of complex biological
systems is carried out largely through various levels of interaction among genes, their products,
and small molecules. To make sense of this complexity, researchers have begun to arrange these
components into varied kinds of biological networks, such as metabolic networks 2, genetic
networks3, protein-protein interaction networks4,5, and genetic interaction networks6 – to put the
pieces back together. A comprehensive knowledge of a cell requires understanding these
biological networks. Although a biological network is not a new concept, a biological network
has many unique features and poses a significant challenge because of its scale and
unprecedented complexity7; therefore, it is important to have a novel conceptual framework to
describe quantitatively a network’s properties as well as new methods for integrating
experimental data and theories.
One important class of biological networks is a genetic network. In a genetic network
each gene and its products are represented, as well as the metabolic steps carried out by
the gene and its product. The flow of activity is built around the Central Dogma and
linked into relevant metabolic activities. Genetic networks are generally circumscribed
by focusing on some particular process, such as sugar metabolism. Such a genetic
network lists all genes and their products thought to be relevant to this particular process
and diagrams how a list of chemical reactions involving genes, their products, and small
molecules explain the process, whether it be sugar metabolism or apoptosis. Thus, genes
and their products together with the small molecules they operate on or with define a chemical
reaction network carrying out the activities of the cell.
To unravel organizing principles and dynamical behavior of genetic networks, formal
mathematical methods for the representation and analysis of genetic networks are required 8. It
would be useful if an analysis of a genetic network could be used to generate testable predictions
as is done in physics9. Linear circuits are widely used models in electrical systems. In the past
several years, metabolic network researchers have attempted to use some concepts of linear
circuits for developing network-based metabolic pathway analyses29. Essential tasks of the
network-based metabolic pathway analysis are to identify the interplay between underlying
structures of the metabolic network and metabolic flux2. A key tool for determining the metabolic
flux distribution is reaction stoichiometry and flux balance equations. A flux balance equation
specifies the mass balance for each metabolite. The flux balance equations can be thought of as
Kirchhoff’s First Law in the context of metabolic networks11-15.
It would be very useful to extend such a circuit analysis to genetic networks.
Applications of circuit theory to genetic networks allow us to develop a unified modeling
framework for both metabolic and genetic networks. Developing a unified model for both
genetic and metabolic networks is desirable for several reasons. First, a unified model
will facilitate integration of different sources of data, such as RNA and protein profiling
information, protein-protein interactions, and metabolic pathway information. The focus
on a genetic network brings together a component, which a geneticist tends to focus on,
namely regulation with the piece that a biochemist tends to focus on, namely metabolism.
Such a unified approach is only beginning to be attempted in a few model systems, such
as the GAL cluster16, qa cluster17, and the lac operon18.
Second, traditionally, genetic, protein and metabolic networks have been
investigated separately. Such a strategy for network study will ignore common rules
underlying structure and function of biological networks. Furthermore, some good tools
that are successfully applied to one type of network may prove useful for other kinds of
biological networks as well. The framework proposed is now described.
The essential feature of a linear circuit is to assume: outputs are linear functions of
the inputs19. Although the relationships between the concentrations of the genes and their
products in the networks are nonlinear20-22, as a first order approximation, linear models
can be used to describe the genetic networks23,24. Kirchhoff’s Current Law and Ohm’s
Law are fundamental to electrical circuits. Applications of linear circuit theory to genetic
networks might proceed by developing analogs to Kirchhoff’s Current Law and Ohm’s
Law in a genetic network. To accomplish these tasks, in the paper, we first present
concepts of transcriptional potential, current and resistance. Then, we develop Ohm’s law
and Kirchhoff’s law for genetic networks. In electrical systems, the resistance can be
experimentally measured. The experimental technologies for measuring transcriptional
resistance in genetic networks have not been developed. However, as an alternative
approach transcriptional resistance could be calculated from regulatory coefficient
matrices in the structural equation models proposed herein for genetic networks. To make
the paper self contained, we will briefly introduce structural equation models for genetic
networks24. After regulatory coefficient matrices in the structural equation models are
found, we then develop algorithms to calculate transcriptional resistance from the
regulatory coefficient matrices. Finally, to illustrate the utility of generalized circuit
analysis of genetic networks, the proposed models and algorithms will be applied to a
part of an apoptosis network.
Ohm’s Law
Resistance is a property that can be characterized by the same expression for very
different systems25. In electrical systems, the concept of electrical resistance is
determined by Ohm’s Law:
R
V
,
I
where V denotes the voltage across a resistor and I denotes the current flowing through
the resistor. The constant of proportionality is the resistance value of the resistor. The
circuit symbol for the resistor is shown in Figure 1. In mechanical systems, the concept of
a viscous friction coefficient is similar to the electrical resistance17. In a translational
damper as shown in Figure 2, the system consists of a piston and an oil-filled cylinder.
The oil damps the relative motion between the piston head and the cylinder. The external
force F acting on the damper is proportional to the velocity difference x of both ends, or
F  bx  b( x1  x2 ) ,
where b is referred to as a viscous friction coefficient.
Although voltage and force are different physical variables, they can be viewed as a
measure of “driving effort”. This view provides an avenue to introduce resistance into the
study of genetic networks. The mechanism of transcription can be briefly described as
follows. Transcription starts with binding of a transcription initiation complex to the
DNA in the promoter region of a gene, which in turn allows RNA polymerase to bind to
the DNA upstream of the coding region of the gene. The result is a transcriptional
complex or transcriptosome that separates the two strands of DNA and forms an open
complex. The open complex moves along one strand of the DNA and transcribes the gene
into mRNA.
The rate of transcription depends in part on the concentration of enzymes, which
activate or repress transcription and on the transcriptosome27. Transcription factors
together with the transcriptosome are driving forces of transcription. The concentration of
a gene can be viewed as a measure of “effort” in performing transcription of a particular
gene. Therefore, the concentration of the gene is defined as a transcriptional potential that
affects the rate of a gene’s transcription and is denoted by P. A transcriptional machine
that involves the whole transcriptional process is referred to as a transcriptor. We imagine
a transcription process as a flow of a gene passing through the transcriptor. The putative
flow of a transcribed gene passing through the transcriptor is defined as the
transcriptional current, which should satisfy Kirchhoff’s first law, and denoted by I. The
relationship between the transcriptional potential and transcriptional current can be
described as a nonlinear algebraic equation:
f(P,I) = 0,
which can be approximated by a linear algebraic equation:
P-RI = 0 .
(1)
The constant R is referred to as the resistance of a linear transcriptor. Equation (1) is a
description of the linear relationship between the transcriptional current and
transcriptional potential and is analogous to Ohm’s law (but for a genetic network).
Kirchhoff’s First Law
In an electrical system, a circuit is specified by its elements and the connection
between the elements. Kirchhoff’s First Law defines how the interconnection of elements
constrains the current flowing through the elements. In an electrical circuit, Kirchhoff’s
First Law states that the sum of all the currents entering a node is equal to the sum of all
the currents leaving a node26. In applying Kirchhoff’s First Law, we need to specify that
currents entering a node should be positive and that currents leaving a node should be
negative.
Kirchhoff’s first law can be generalized to genetic networks28. At steady-state in
the reactants and products of a genetic network, the expression levels of genes in the
network can be described by the following linear system of algebraic equations:
AX=0
(2)
The structure of the network can be mathematically described by a node-edge incidence
matrix. The column in the matrix associated with edge (i, j ) contains a “-1” in row i, a
“+1” in row j, and zeros elsewhere. If the edge is connected to only one node, then the
corresponding column in the matrix contains only “-1” or “+1”, depending on that the
edge exits or enters the node, respectively. Let B be the node-edge incidence matrix. If
we assume that the rank of the matrix B is equal to the number of the genes in the
network, then we can decompose the matrix A into
A  BY
(3)
Let
I=YX .
Then, the equation (2) is reduced to
BI = 0
(4)
Equation (4) mathematically defines Kirchhoff’s First Law in genetic networks. It
implies that the algebraic sum of all transcriptional currents entering and leaving a node
is zero. Kirchhoff’s First Law may be expressed as
K
I
i 1
i
 0,
(5)
where Ii is the i-th current entering (or leaving) the node and k is the number of the
total currents. To illustrate Kirchhoff’s First Law, we present an example of a genetic
network which is a part of an apoptosis genetic network28 as shown in Figure 3. In
Figure 4, the number in brackets is an index of the node. For each gene, we can write the
transcriptional current balance equations as follows:
I1  I 2  I 3  0
I2  I4  0
I3  I5  I6  0
I6  I7  0
I 7  I8  0
I8  I9  0
I 4  I 5  I 9  I10  0
The node-edge incidence matrix B is given by
1  1  1 0 0 0 0 0 0 0 
0 1
0  1 0 0 0 0 0 0 

0 0 1
0 1 1 0 0 0 0 

0
0
0
0 0 1 1 0 0 0 
B


0 0 0 0 0 0 1  1 0 0 
0 0 0 0 0 0 0 1  1 0 


0 0 0 1  1
0 0 0 1 1
Transcriptional conductance and its calculation
Another useful quantity in circuit analysis of genetic networks is known as the
transcriptional conductance, defined as
G
1
.
R
Ohm’s Law can be rewritten in terms of the conductance as
I=GP
Transcriptional resistance and conductance may be theoretical concepts at the moment.
Unlike resistance and conductance in the electric circuit, which can be experimentally
measured in the electrical system, transcriptional resistance and conductance are difficult
to measure, but they could be calculated from structural equation models for genetic
networks rather than directly measured by experiments. For the convenience of
presentation, we assume that the graph describing genetic networks is a directed acyclic
graph. Suppose that a genetic network is described by the following structural equations:
Pi 

jE1 ( i )
aij Pj 

lE 2 ( i )
il
X l , i  1,..., n ,
(6)
where E1(i) is the set of total neighboring nodes (endogenous variables) of the node i with
the edges directed to the node i and E2(i) is the set of total neighboring nodes (exogenous
variables) of the node i with the edges directed to the node i, Pi is the concentration of the
i-th gene in the network, Xl is the l-th exogenous variable, constants aij and γil are the
coefficients in the structural equations.
Suppose that the k-th path leaving the node i (endogenous variable) consists of a
sequence of the nodes n1n2 ...nvi , where the node n1 lies on the boundary of the network
and all the nodes are endogenous variables. Let H i (k )  a n1n2 a n2 n3 ...a nvi and L(i) be the
set of the neighboring nodes of the node i with the edges leaving the node i. The k-th
path leaving the node i can be partitioned into two parts: n1n2 ...nv and nv i . Let Gij be the
transcriptional conductance of the edge between the node i (endogenous variable) and the
node j (endogenous variable), and Yil be the transcriptional conductance between the node
i and the node l (exogenous variable) . Then, we have
Gij 
Yil 
 H (k )a
i
k L ( i )
 H (k )
k L ( i )
i
ij
(7)
il
If the node i is at the boundary, then Hi(k) =1.
To illustrate how to calculate the transcriptional conductance, given the regulatory
coefficient matrix, we applied the proposed algorithms to TGF-β pathways which are
shown in Figure 4. The numbers in the brackets are regulatory coefficients in the
structural equations. The structural equations for TGF-β pathways are given by
P3  0.1753P1  0.1390 P2
P6  0.6879 P3  0.5696 P4  1.3643P5
P7  0.1211P4  0.4738P5
P8  0.2013P2  0.1352 P5
P9  0.9977 P6
P10  0.5607 P6  0.4372 P7
Since the nodes Collagen XI (9) and Collagen III (10) are at the boundary, by the
equation (7), we have
G10, 6  0.5607, G10, 7  0.4372, G9, 6  0.9977
For the node CTGF (6), there are two paths leaving the node CTGF (6):
CTGF -> Collagen XI and Collagen I -> Collagen III.
Thus,
H 6 (1)  0.5607, H 6 (2)  0.9977.
Using two quantities H6 (1) and H6 (2), we have
G6,3  (0.5607  0.9977)  (0.6879)  1.0720
G6, 4  (0.5607  0.9977)  (0.5696)  0.8877
G6,5  (0.5607  0.9977)  (1.3643)  2.1261,
where the negative conductance means the change of direction of the current. Other
transcriptional conductances are showed in Figure 4 (numbers outside brackets).
Nodal analysis
Ohm’s Law is fundamental to circuit analysis. However, even the simplest circuit
requires application of Kirchhoff’s Law. Circuit analysis is based on constraints of two
types: (1) connection constraints (Kirchhoff’s law) and (2) device constraints (Ohm’s
law). In nodal analysis, the node (endogenous variable) potential is assumed unknown.
Once node potentials are found, the transcriptional currents can be obtained by Ohm’s
Law. For a circuit containing N nodes (endogenous variables), there are N equations,
which are determined by Kirchhoff’s First Law, with N unknown node potentials. N
equations with N unknown variables may be simultaneously solved.
The application of Kirchhoff’s First Law at a node, expressing each unknown
transcriptional current in terms of the node transcriptional potential, results in a node
equation. The network in Figure 5 will illustrate the formulation of a node equation.
To solve the equations, we first need to assume that independent sources have values
that are given. In Figure 5, we assume that the potential P1 is known. The application of
Kirchhoff’s First Law leads to the following two independent equations:
 I12  I 23  I 24  0
 I13  I 23  I 34  0
(12)
Using Ohm’s Law, we express the transcriptional currents in terms of the transcriptional
potentials as follows:
I12  G12 P1 , I13  G13P1 , I 23  G23P2
I 24  G24 P2 , I 34  G34 P3
(13)
Substituting equation (13) into equation (12) yields
G23P2  G24 P2  G12 P1
 G23P2  G34 P3  G13P1
Solving the above equations results in
P2 
G12
P1
G23  G24
P3 
1
G G
(G13  23 12 ) P1
G34
G23  G24
Similar to the reference node in an electrical circuit analysis, where the node voltages are
then defined as the voltage between the remaining nodes and the selected reference node,
we choose node 4 as an undetermined node in the network circuit analysis, where the
transcriptional potential P4 is assumed to be undetermined.
Discussion
Cell function can be understood in terms of the operation of biological networks.
Although systems biology has significant momentum, it is still elusive in terms of its
success stories35,36. To uncover the principles governing the operation of biological
networks is a key challenge for systems biology.
An important problem for systems biology is to develop quantities which describe the
operation of genetic networks and which measure the behavior of these networks.
Concepts of potential, current, and resistance historically have played an extremely
important role in electrical circuit analysis. Current (or flux, or flow) can describe and
measure movement in materials. Potential (or voltage) quantifies the “driving force” or
the degree of causation. Conductance of an element measures the effect of the “driving
force” on the element, which determines the distribution of the currents. Kirchhoff’s Law
and Ohm’s Law are principles underlying the functional relations of the potential, current
and resistance (or conductance).
Generalizing the concepts of the potential, current and resistance, and Kirchhoff’s
Law and Ohm’s Law to genetic networks has the following advantages. First, they are
linked to a simple conceptual model of genetic network under study and provide a
framework for describing the operations of a genetic network. Second, the potential,
current and resistance explicitly define the major variables in the network and hence can
be used to quantify the operation of a genetic network. Third, Kirchhoff’s First and
Ohm’s Laws capture relationships among the variables and processes that comprise the
genetic networks and hence are principles governing the operation of genetic networks.
Fourth, tools for circuit analysis can be used to make testable predictions about how
genetic networks will behave. Given inputs to the networks, the operations of the
networks can be predicted. The accuracy of the predictions can be evaluated by
experiments. It follows that circuit theory will help us to formalize our understanding
about the various genes and their products explaining a particular biological process, such
as apoptosis. The formalized hypotheses are then transformed into circuit equations. The
accuracy of prediction of the variables and parameters in the equations can be used to
evaluate how well the data support the hypothesis. Finally, the generalized circuit theory
in the genetic network analysis here specifies the relationships between inputs and
outputs and allows consideration to designing a biological system which will better
respond to environmental changes, such as the introduction of anti-cancer drug
candidates. Therefore, the circuit model of the genetic networks may find broad
applications in drug development.
In this paper, we have generalized linear circuit theory in electrical system analysis to
genetic network analysis by first introducing concepts of transcriptional potential, current
and resistance. Then, we developed Ohm’s Law and Kirchhoff’s law for genetic networks.
Technologies for measuring resistance of the transcript have not been developed. As an
alternative, we have developed algorithms to estimate transcriptional resistance based on
structural equation models for genetic networks. The key issue for application of linear
circuit theory to genetic network analysis is the assumption of linear models for genetic
networks. The precision of linear circuit analysis of genetic networks depends on (1) the
accuracy of the structure of the networks and (2) how well the linear structural equations
model the behavior of the genetic networks. As advancement of the modern biology,
available knowledge of network structure will continue to grow. Under specified cell
conditions, the nonlinear relations between the transcriptional potential and current can
be approximated by linear model near the operating point of the genetic networks. The
transcriptional resistance depends on the magnitude of the currents and the environmental
parameters. Their relationships should be further investigated. In this paper, we only take
first step to initiate development of the generalized circuit analysis of genetic networks.
Deeper investigation toward this direction should be carried out in the future.
Acknowledgements
M. M. Xiong is
supported by NIH-NIAMS Grant IP50AR44888 and NIH Grant
ES09912.
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Figure Legend
Figure 1. Circuit symbol for the resistor.
Figure 2. Translational damper.
Figure 3. Transcriptional currents in a part of an apoptosis genetic network.
Figure 4. Regulatory coefficients and transcriptional conductance in TGF-β pathway,
where the number enclosed by bracket was regulatory coefficient in the structural
equations and the number, which was not enclosed by the bracket, was the transcriptional
conductance.
Figure 5. Scheme of transcriptional circuit.
Figure 1 Circuit symbol for the resistor.
Figure 2. Translational damper.
Figure 3. Transcriptional currents in a part of an apoptosis genetic network.
Figure 4. Regulatory coefficients and transcriptional conductance in TGF-β pathway,
where the number enclosed by bracket was regulatory coefficient in the structural
equations and the number, which was not enclosed by the bracket, was the transcriptional
conductance.
Figure 5. Scheme of transcriptional circuit.