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Í VOHRADSK
Institute of Microbiology, CAS,142 20 Prague, Czech Republic
1Correspondence: Institute of Microbiology, CAS, Vide
ská 1083, CZ-14220 Prague, Czech Republic. E-mail: vohr{at}biomed.cas.cz
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODELRESULTSDISCUSSIONREFERENCES |
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, J. Neural
network model of gene expression.
Key Words: genetic network • regulation of gene expression • mathematical modeling
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONREFERENCES |
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Transcriptional regulation is conferred through the action of gene products that bind to each gene’s transcriptional start site. These proteins bind directly to DNA and influence gene expression by altering the activity of transcriptional machinery. The activity of transcription factors is controlled by other gene products. Thus, the transcription of a particular gene can be viewed as a combinatorial action of products of other genes or even of its own gene product. Once the expressed gene is translated into a functional gene product, it affects the state of the cell and may directly or indirectly influence the expression of other genes. In this way the expression state of the cell is controlled by a set of genes expressed at a given moment. These genes regulate the transcription of cell genes leading to a new state. The new state is defined by the concentration of currently expressed gene products, and so on. This means that the state of transcription and translation in the cell is continuously updated from one state to another in a feed-forward-like circuit, where the current state depends on the previous one, which depends on the preceding one, etc.
This approach can be formalized as a concept of ‘genetic network’
where all genes and their products participating in one regulatory
event are members of a network. The transcriptional state of a gene is
defined, in principle, by the states of all other genes in the network
(Fig. 1
). This model can be simplified by defining the expression of genes as
either on or off {0,1} (1
2
3)
. The response of a gene
in the model is then defined by Boolean rules that combine the states
of expression of genes controlling the particular gene (e.g., gene A is
expressed if the formula C AND (B OR D) for genes B, C, D is true).
When the Boolean rules for all members of the model are known, the
terminal state of the network can be computed. The control of gene
expression can thus be simplified to wiring between genes and a set of
Boolean rules associated with the synapses (using neural network
terminology) connecting the nodes. The synapses and the rules define
the next state of the network. From the theory of random Boolean
networks, it follows that the final state of the network (the state
after a finite number of state updates) is defined by either a point
attractor or a basin attraction (1
, 4)
. A point attractor
defines a stable state that does not change with the update of the
network. The basin of attraction represents a set comprising a limited
number of regularly changing states (limited cycle). Several methods to
identify the network from experimental data have been suggested
(3
, 5)
.
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This Boolean simplification ignores the genes that have different
regulatory effects depending on their level of expression. Furthermore,
these networks cannot address those genes that influence the
transcription of various genes to different degrees. The model has
therefore been generalized to incorporate different levels of
expression of regulatory genes and different degrees of the regulatory
effect on a given gene. D’Haesleer (6)
suggested a linear
model of gene control where the expression level of a gene is defined
by the linear combination of expression levels of other genes weighted
by a value reflecting the level of their regulatory effect. The weight
matrix then summarizes the interactions among all members of the set.
The methods for computing the weight matrix from experimentally
measured expression time series are presented. The authors also discuss
limits of the approach based on a linear model that oversimplifies the
expression kinetics. Weaver et al. (7)
extended this
approach by introducing a nonlinear ‘squashing function’ modifying
the output from the linear combination of weights and expression levels
of the regulatory genes. The additive principle for updating the state
of the network was kept. The method of reconstruction of the weight
matrix from experimental data was also analyzed. Genetic networks have
also been modeled by a set of differential equations (8)
.
Each such model should be able not only to fit the experimentally measured expression time series ‘profiles’, but also to correctly describe the underlying process. As the weight matrix has to be derived from the experimental data according to a certain model, an invalid model will produce an invalid matrix and hence an invalid reconstruction of the regulatory connections. Unfortunately, little (if any) experimental data exist that would allow the model to be tested. Nevertheless, rapid developments in DNA chip technology and quantitative proteomics that can provide such data promise to fill this gap. In the current situation, the model can be tested only on simplified artificial examples where the output can be estimated.
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MODEL |
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TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONREFERENCES |
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, where each node represents a particular gene
and the wiring between the nodes define regulatory interactions. Each
layer of the network represents the level of expression of genes at
time t. Output of a node at time t+
t can be derived from the
expression levels at the time t (yj) and
connection weights (wij) of all genes
connected to the given gene. Thus gi, a
regulatory effect to gene i, is
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Let the rate of expression of gene i be given by the
regulatory effects of other genes
i and the effect of degradation
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i represents the regulatory effect
reflected in a variable gi
(i =
k1igi). The
constant k2 represents the rate constant
of degradation of the gene product i and
k1 is its maximal rate of expression. The
whole model has the form:
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represents a special case of a class of recurrent
neural networks that can be described by the general formula
(9)
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is an external
input to the node i. These types of networks have been used
as associative memories and as models of brain activity, and their
dynamics have been thoroughly studied. The output of the network at
time t depends on the weight matrix w. If the
matrix is symmetric (wij =
wji) the network reaches a stationary state in
finite time. If the connection weight matrix is far from being
symmetric, the state transition of the network can lead to a point
attractor (stationary state) or can become oscillatory (limit cycle) or
even chaotic, depending on the complexity of the network. |
RESULTS |
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TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONREFERENCES |
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(Fig. 2
). Here the product of gene A controls the expression of gene B, which
initiates the expression of gene C. Gene B induces the expression of
gene A forming positive feedback. Gene C in turn negatively controls
the expression of gene A, forming negative feedback. This is a simple
feedback circuit that can be characterized by a network and a weight
matrix (Fig. 2)
. Considering only one isolated circuit, the state of
the network can reach two attractors, depending on the initial
conditions: a point attractor, where the action of the network is
stopped; or a two-point attractor perpetually oscillating between two
states.
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In the case of multiple copies of the circuit, the reaction of the system depends not only on the state of the genes (on/off), but also on the concentration of each compound controlling the process. It is evident that with increasing concentration of the product of A, expression of B will increase with a delay proportional to the reaction kinetics of the transcriptional/translational machinery of gene B. This is also valid for the B/C pair. Therefore, an increased concentration of the A product will be followed by a delayed increase of B and, with another delay, by an increase of C. This scheme will be maintained until the concentration of C product is high enough to block the synthesis of A, reflected in a decrease of the concentration of A, since all products are continuously being degraded. In turn, a decreasing A directly causes a decrease of B and consequently of C. If the concentration of A or B is close to 0, the whole process is terminated. If not, after the decrease of C, A is synthesized again; the whole cycle starts up again and is repeated ad infinitum.
This is a qualitative analysis of the behavior of the model of
expression shown in Fig. 2
. We have reconstructed the response of the
network from Fig. 2
. using the following weight matrix
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,
and solved the system of equations numerically. The results are shown
in Fig. 3
.
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The parameter b (Eq. 2)
represents the delay with which the
transcription/translation process of one gene reacts to the regulatory
impulse from other gene products. High negative values mean a long
delay. If all parameters are set to the same values (Fig. 3a
, c
), the system terminates at a point attractor. If the reaction of
one component is faster, e.g., gene A (Fig. 3b
), then after
the concentration of the C product (which blocks transcription of gene
A) is low enough, transcription from gene A is quickly renewed and the
cycle starts up again. The system terminates in a limit cycle: the
expression levels of all components oscillate with time. All other
parameters controlling the rate of degradation and expression were kept
at 1.
Connected networks
Let us combine two networks by introducing a connection between C
of the first network and A of the second one (Fig. 4
). We would expect that the output of network 2 would be delayed,
waiting for a sufficient concentration of the C product. If the weight
matrix for the second network is the same, the response of the second
network would be delayed but otherwise qualitatively identical to that
of network 1. This is exactly what we observe in the kinetic profiles
of Fig. 4
.
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Linked networks: transcriptional and translational control of gene
expression
The expression of a gene in the previous examples is influenced by
the regulatory effect of other proteins involved in the regulatory
event, and the rate of protein formation was replaced by a nonspecific
sigmoid curve representing both translation and transcription. If we
separate the two processes (Fig. 5
), transcription and translation can be considered as controlled in two
separated compartments. This concept was originally developed by Hoel
(10)
and Yagil (11)
and extended by Hargrowe
and Schmidt (12)
(readers interested in the validity of
the assumptions used in the model should examine these references and
their bibliographies). The process can be described by two different
networks connected by one synapse. The connection is maintained by the
mRNA of the specific gene. If we consider that synthesis of mRNA
corresponding to gene i (ri) is controlled
by the regulatory proteins A... n (Fig. 5)
and the rate of its
synthesis again follows a sigmoid, the rate of mRNA accumulation will
be described by the equation:
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) ‘squashing’ the influence of
proteins p using the weight matrix
wr.
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The rate of protein formation in the process of translation is
controlled by the mRNA concentration ri
and the action of other molecules
(
j) involved in the translational
process. The following equation describes the rate of accumulation of
protein pi:
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using new
weight matrix wp. Protein
pi then can, in principle, contribute to the
regulation of the expression of other genes of the system (Fig. 5)
forming a feedback circuit.
Equations 7
and 8
form a ‘two-compartment’ model of gene expression.
The two networks are connected by one synapse formed by
mRNAi. Both networks can be evaluated from
experimentally measured mRNA and protein concentrations. Equations 7
and 8
can be written for all n members of the regulatory
process forming a system of 2n differential equations.
The translation of an mRNA to the corresponding protein is a
several-step process. It has been considered that the components of the
system are available in excess. Therefore, the rate of protein
accumulation is determined only by the amount of mRNA. The weight
matrix thus becomes zero and equation 8
simplifies to:
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).
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We can consider that the rate-limiting step is not only constrained by
the mRNA concentration, but by one or more of the processes of the
translational machinery. In this case, the elements of the weight
matrix wp gain non-zero values. In
principle, the time series of mRNA and protein can differ. This has
been observed previously and recently shown experimentally (13
, 14)
. The simulation of such situation is shown in Fig. 7
.
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Reconstruction of the network architecture from experimental data
The network architecture is uniquely determined by the number of
nodes and the wiring between the nodes. The wiring is defined by the
weight matrix. The number of nodes is defined as the number of genes or
other factors involved in the regulatory event. Ultimate reconstruction
of the network therefore means computation of the weight matrix from
the experimental data. The only data available are the time series of
expression levels of the genes involved in a particular regulated
system. We can measure or at least estimate the decay rate constant
(k2) and the maximal rate of expression
(k1) (the decay rate constant can be expressed in
terms of a proteins half-life t1/2, which is
related to the k2 constant by the formula
t1/2=ln2/k2).
For every recurrent network, there exists a feed-forward network
equivalent to the recurrent network over a finite time period
(15)
. The basis of the strategy is the multiplication of
the units and connections of the recurrent network over limited time.
In fact, this unfolding is equivalent to the situation shown in Fig. 1
.
The state of the network at an arbitrary time
t+t is given by the state at the time
t and the transition is described by a set of simultaneous
differential equations such as those described by equation 4
.
Provided that the network relaxes to a steady state, recurrent
back-propagation (16)
can be used to reconstruct the
weight matrix from known initial and terminal conditions. For a general
type of response, stochastic optimization like simulated annealing
(17)
, which is capable of finding the global minimum in
the parameter space representing the unique solution, has to be
adopted.
Difficulties arise when the system is underdetermined (fewer
experimental data points then equations). Then an infinite number of
solutions of equation 4
exist. Unfortunately, this is the predominant
case; the number of experiments is limited by the physiology of an
organism and by technical and scale constraints. In this case, a
nonlinear interpolation scheme has to be applied to the expression time
series that generates a sufficient number of linearly independent
equations. A danger of this approach lies in the fact that some rapidly
occurring changes in the expression profiles can be hidden between two
measuring points within an interval too long to capture these changes.
In unequally spaced data, an interpolation yielding equal spacing gives
higher weights to points with a longer interval between them. The
general rule is that the sampling intervals should be short enough to
capture at least the principal trends.
Another problem is introduced by linear dependencies between different kinetic profiles of different gene products. This introduces singularities in to the model and makes the solution impossible. Linearly dependent profiles are results of control by the same set of genes but to a different extent (they are coregulated). As the goal is reconstruction of the network structure, these different profiles represent identical wiring patterns and can thus be replaced by one synapse only. Therefore, preprocessing (using, for example, cluster analysis methods to group profiles similar in shape) has to be applied.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONREFERENCES |
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, 2
, 18
, 19
, 20)
; 2)
models based on kinetic equations and binding equilibria (21
, 22
, 23)
or an artificial system of differential equations
(8)
; 3) McAdams introduced an approach that
interprets a genetic control system as a circuit formally similar to an
electronic one (24
, 25
, 26)
; 4) Neural
network-based models that replace gene interactions with a weight
matrix and linear (6)
or sigmoidal (7)
transfer functions and a linear combination of the descriptive
equations.
In principle, cell regulatory processes can be described by a set of
binding equilibria and kinetic differential equations. Such an approach
was used, for instance, in the modeling of a biochemical pathway for
chemotaxis (21
, 22)
. The constants of the model were
obtained from measurements or estimated by fitting the model to the
experimental data. A limited genetic algorithm was used to search for
sets of binding reactions and associated binding constants expected to
give mutant phenotypes in accord with experimental data
(23)
. This approach considers a process in the cell as a
physicochemical system analogous to the in vitro situation.
In such cases, it is assumed that all components of the system and the
interactions between them are known and all necessary constants can be
measured or estimated. A model that fully describes the behavior of the
system can thus be formulated.
In the case of gene expression, such a situation is difficult to achieve as the mechanisms of the processes of gene control are not completely understood. Replacing unknown mechanisms by a kinetic equation with necessarily virtual constants derived from fitting the model to experimental data means the same level of abstraction as replacement of the process by an artificial model like the one presented here, but still with the rigid form of a kinetic equation. For poorly identified systems, where one kinetic equation can comprise several processes, the description by kinetic equation is invalid and does not correctly describe the dynamics of the system. Such a situation can occur in a neural network model as well, but our model does not strictly follow the chemical reaction pathway; it is designed to be flexible enough to follow the observed dynamic behavior. For any system that is not completely understood, a dominant case in most biological systems, the weight matrix (derived from the fitting of the model to the data) can at least give a qualitative indication of the principles of regulatory interaction among the elements of the system. The physicochemical type of model needs to know this information a priori. Known experimental errors can provide estimates of the boundaries for the weight matrix elements, i.e., whether the mode of action of one component on another is significantly positive, negative or zero. This minimalist qualitative result can draw the attention of experimental biologist to a limited number of significant components of the system. New experiments that bring new information can then be made and the model can be improved. Such an iterative approach leads to an accurate model that can be tested and can finally provide valid predictions.
The Boolean network model interprets gene interactions as connections between genes and the states of gene expression that can be either on or off. The next state of the network is calculated from the known previous state from the rules for interactions between elements of the network and from the known regulatory connections between the genes. The network scheme interpreting the gene control system as a network is identical to the one presented here. The difference is in the interpretation of the regulatory function. In our case, the model can compute expression levels of mRNAs or proteins at any time, whereas the Boolean network model can suggest whether the gene is on or off during the given cycle. The Boolean model does not consider that even when the gene is off the concentration of the gene products can still be high enough to act in the regulatory process. It maintains gene activity even when the gene is already not transcribed. If this shift occurs early in the process, the error is propagated in time and finally can lead to a different state of the network than the one predicted by the Boolean network model. Simulation of behavior of the simple networks presented here showed that between the two limit cases defined by the Boolean network model, a number of different kinetic and terminal states can exist. These states are determined by the combination of parameters comprising the basic physicochemical and kinetic features of the members of the system, and not only by the initial state of expression and the weight matrix as suggested by the Boolean model. For the same weight matrix (same interactions), different kinetic behaviors and terminal states can be reached depending on the other parameters and initial conditions. It is very likely that the behavior of Boolean networks modeling a regulatory event represent a limiting case that can be reached by the system, but not inevitably. Other solutions leading to different terminal states of the system starting from the same initial conditions are also possible. Particular initial conditions can lead to attractors different from those suggested by the Boolean networks. Therefore, the neural network model is more realistic and also more accurate, providing information about instantaneous concentrations of the components of the system, not only about the state of genes transcription. Reconstruction of the Boolean network from the experimental data that are expected to be available, which means mostly DNA chip and proteomics data, would also be difficult.
Circuit simulation of genetic control has a great advantage in the
apparent analogy between genetic and electronic circuits and the
existence of a broad theoretical background coming from electronic
engineering. The circuit-based models may lead to a robust
interpretation of complex regulatory networks, but the question
remains, how valid is the replacement of gene regulation processes by
discrete electronic-like elements? An approach using fuzzy elements
(27)
seems to improve the validity of this type of model.
Reconstruction of the circuit from the experimental data will probably
be another challenge.
The advantage of the model presented in this paper is that it is continuous in time, does not use artificial elements, and uses a transfer function that transforms the input to a shape close to the one observed in natural processes. The model allows us not only to fit the experimental data, but can also provide information about the principles of control and mutual interactions of the elements of the modeled system even when only the elements are known in advance, and nothing about their interactions. Furthermore, the model can, from known principles of interaction of the elements of the system, compute the dynamic behavior of the system and the possible terminal states the system can reach. Predictions made by such a model outside the measured interval are likely to be more accurate than those made by totally artificial models.
The model is open for modification. It works with concentrations (amounts) and calculates the instantaneous concentration of the elements of the system by means of the set of equations described above. If there exists a known mechanism of control of the concentration of one of the elements of the system, the equations controlling these elements in the model can be replaced by the new ones. Such a hybrid model has a higher chance to be the correct description of the system than a purely artificial one.
In our model, we consider gene expression as a two-stage process of transcription and translation, as in the real cell. All the previously published models we were able to find consider the process as one stage, ignoring differences in the control of translation and transcription.
The drawback of the proposed model is that it still comprises a large number of parameters that have to be computed from experimental data. Many parameters require large number of data points to correctly compute the parameters. The number of data points obtained from experimental time series is limited by technical constraints that can hardly be overcome. Therefore, interpolation schemes and minimization procedures that bring inaccuracy to the results have to be applied.
Solution of the differential equations representing the model to obtain the weight matrix has to be accomplished numerically. The schemes mentioned above, which in principle search for a minimum in a parametric space, never guarantee that in any particular case the global minimum representing the correct solution will be reached. Therefore, for the best possible results it is necessary to introduce constraints coming from experimental observations and known features of the modeled system. It implies that building up a successful model will be an iterative process involving both mathematicians and experimental biologists. This kind of collaboration is inevitable for any kind of modeling approach that has been or will be invented.
Another simplification in our approach is that the model presumes that
all necessary molecules are available as in an in vitro
solution. Therefore, terms representing the diffusion of molecules to
the place where the reaction takes place or compartmentalization
(28)
of the cell where all necessary agents are pooled
should be considered. Other environmental variables can be included to
describe a particular problem. On the other hand, compared to the
previously published ones, this model already comprises a nonlinear
response to the regulatory action of the other genes of the system, the
effect of degradation of the gene product, different rates of
expression of individual genes, and their regulatory effect. How much
these terms are sensitive to random perturbation or experimental error
could be tested with data obtained from a known system with a known
experimental error. A series of similar models can be constructed by
distributing the input data within the known standard deviation for
each measurement. Comparing these models would tell us how sensitive
the models are with respect to small amounts of noise in the input
data.
Once a sufficiently comprehensive model is established, its dynamic
properties can be tested by mutageneses in vivo and ‘in
silico’ and results compared. This is a general way of refining the
functionality of the model. The advantage of a computer-based model is
that it can predict situations that can hardly be reached
experimentally and can explain underlying mechanisms that are, in
principle, inaccessible. Even analysis of the very simple model shown
in Fig. 2
demonstrates very complex behavior, depending on the
combination of the parameters of the model. Randomly generated
expression profiles for the same weight matrix range from a complete
stop of expression through oscillatory behavior to saturation. It shows
how narrow the range of biologically plausible parameters is and how
tight control must be in the living cell.
Results show that the model presented here reproduces well the predicted behavior of these simple artificial examples. By playing around with the values of the parameters, various model situations can be tested. I ran simulated time-course experiments for large networks (up to 100 nodes) with randomly generated weight matrices and parameters. These networks were allowed to iterate until they reached a steady state or a cycle. The dynamics of the network output depended on all parameters of the model and the initial conditions. All networks were generated with randomly selected average connectivity (number of connections per node). Not surprisingly, large networks required a longer period of time to reach a terminal state.
These observations emphasize how finely tuned and tight the regulatory system of the cell must be in order to be flexible enough to adapt to changing environmental conditions and yet maintain global stability. It is encouraging that the neural network approach can reflect this complexity, indicating that it is a good base for modeling of such complex regulated systems.
Although these models include idealization of known features of gene control, they are open to the addition of other different modes of interaction and responses and are already capable of describing simple regulatory events occurring in the cell. To what degree these models are real pictures of reality remains to be proved by comparison with experimental data of the known systems.
The cell as a parallel processing network with a limited number of
states
Bray (29)
discussed an interpretation of the
enzymatic reaction as a memory unit capable of transforming an input
(concentration of substrate) to an output (type and concentration of
product). The combination of such nodes can form a network that can
serve as a decision block adapting its output to its input. The output
can be both extremely complex and simple [for example, the decision
about the direction of rotation of a bacterial flagellum (22
, 28)
]. McAdams and Shapiro (24)
used the analogy of
an electronic circuit to explain the process of decision for the
lysogenic or lytic path after phage lambda infection. Both examples
consider biochemical reactions as logical (or, more generally, fuzzy
logical) units capable of storing and processing information. In our
case, basic processes of transcription and translation are modeled by a
set of equations principally identical to the artificial recurrent
neural networks. Recurrent neural networks have often been used as
associative memories. Networks can be trained to reach, after a certain
time, some finite number of states depending on the input to the
network. When the input is changed, the output state of the network
relaxes to another of the possible states: the network can ‘remember’
the input. This approach leads to the concept of ‘black box’, where
the only required function of the unit is transformation of an input to
the output in the observed way. Neural networks are excellent tools for
the construction of this architecture. In such a case the weight matrix
does not necessarily have to have a direct meaning in terms of
influence of certain known parameters, but can be artificial, serving
only for the purpose of correct transformation of the input value.
To get the correct concentration of necessary products at a certain time, gene expression is controlled in a complex way, reacting to changes in the developmental program or environmental conditions. The cell is allowed to reach only a limited number of states that maintain its stability. Therefore, a large number of possible input conditions can lead to only a limited number of terminal states over time. The existence of a restricted number of allowed states maintains the regulatory stability of the system on one hand and, on the other, its variability maintains the necessary flexibility. The situation resembles the attractors of an artificial neural network and suggests that the neural network could be an acceptable model of control of cell processes in general. From this point of view cell control can be considered to be a system of mutually connected networks processing in parallel. Whether this model is plausible and to what level this generalization is valid will be the subject of the future studies.
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ACKNOWLEDGMENTS |
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Received for publication June 26, 2000.
Revision received September 15, 2000.
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REFERENCES |
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... 
. Proteini can control
expression of itself or other genes. Such a scheme can be drawn for any
of the members A... n. The formal description of the process is
shown in equations 7
