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ABSTRACT In cellular regulatory networks, genetic ac- tivity is controlled by molecular signals that determine when and how often a given gene is transcribed. In genetically controlled pathways, the protein product encoded by one gene often regulates expression of other genes. The time delay, after activation of the first promoter, to reach an effective level to control the next promoter depends on the rate of protein accumulation. We have analyzed the chemical reactions con- trolling transcript initiation and translation termination in a single such ‘‘genetically coupled’’ link as a precursor to modeling networks constructed from many such links. Sim- ulation of the processes of gene expression shows that proteins are produced from an activated promoter in short bursts of variable numbers of proteins that occur at random time intervals. As a result, there can be large differences in the time between successive events in regulatory cascades across a cell population. In addition, the random pattern of expression of competitive effectors can produce probabilistic outcomes in switching mechanisms that select between alternative regu- latory paths. The result can be a partitioning of the cell population into different phenotypes as the cells follow dif- ferent paths. There are numerous unexplained examples of phenotypic variations in isogenic populations of both pro- karyotic and eukaryotic cells that may be the result of these stochastic gene expression mechanisms.
In all organisms, networks of coupled biochemical reactions and feedback signals organize developmental pathways, me- tabolism, and progression through the cell cycle. For example, overall coordination of the cell cycle results from an overar- ching set of dependent pathways in which the initiation of late events is dependent on the earlier events and the whole operates as a form of biochemical machine. Within these regulatory networks, genetic activity is controlled by molecular signals that determine when and how often a given gene is transcribed. Additional signals stimulated by environmental influences or by signals from other cells can affect the ongoing reactions to influence the future course of cellular events. Since a regulatory protein may act in combination with other signals to control many other genes, complex branching net- works of interactions are possible. In these nets, one regulatory protein can control genes that produce other regulators, that in turn control still other genes.
How long does it take for these messages and controlling influences to move through a regulatory cascade? In biochem- ical regulatory networks, the time intervals between successive events are determined by the inevitable delays while signal molecule concentrations either accumulate or decline. Genet- ically coupled links are links where the protein product en-
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Copyright 􏰁 1997 by THE NATIONAL ACADEMY OF SCIENCES OF THE USA 0027-8424􏰀97􏰀94814-6$2.00􏰀0
PNAS is available online at http:􏰀􏰀www.pnas.org.
coded by one gene regulates expression of other genes. The time delay in genetically coupled links (Fig. 1) depends on the time required for protein concentration growth, after pro- moter activation, to the concentration range that controls the next level in the cascade. Conversely, the time delay after the controlling promoter turns off depends on the time for the protein concentration to decay below the effective range. Fig. 1B shows a common architecture for such genetically coupled links. In these links, for appropriate combinations of input signals, transcripts are initiated and the protein product ac- cumulates when production exceeds degradation; the increas- ing protein concentration simply broadcasts the information that the promoter is ‘‘on.’’ The message is ‘‘received’’ or detected by the concentration-dependent response at the protein signal’s site(s) of action, stimulating a response at each site in accord with that site’s chemical behavior. (We use the term ‘‘protein signal’’ to mean the regulatory protein concen- tration in its effective form at its site of action.)
In this paper we examine the properties of a single geneti- cally coupled link as a precursor to modeling networks con- structed from many such links. Specifically, we ask what determines the time required for protein concentration to grow to effective signaling levels after a promoter is activated and how statistical variations in this time can affect observed cellular phenomena across a cell population. It has been proposed that the pattern of protein concentration growth is stochastic, exhibiting short bursts of variable numbers of proteins at varying time intervals (2, 3). (Herein the term ‘‘stochastic’’ is used in the statistical sense of resulting from a random process.) We formalize and quantify this notion of randomness in genetic regulatory mechanisms by explicitly characterizing the statistics of the random processes implicit in the chemical reactions (4). By analogy to electrical circuits, we will refer to this time interval between the switching on of the first promoter and activation or repression of the second promoter as a ‘‘switching delay.’’ There is also a switching delay of a different magnitude for the inverse functions when the controlling promoter is switched off. We are neglecting here the case where multiple molecules act combinatorially to determine the controlling action.
Then, as a concrete illustration of switching delays over a genetically coupled link, we simulate a representative link using parameters characteristic of links in bacterial regulatory networks. The simulation results show that short-term fluctu- ations in protein production can be large relative to signal thresholds that control expression of critical genes. For the same link in different cells of the same genotype, there will be wide random variations in both the times to produce a given protein concentration or in the number of proteins produced when the promoter is transiently activated. Implications of this noisy pattern of gene expression for cellular regulation include: (i) the switching delay for genetically coupled links, hence the time for the cell to execute cascaded functions, can vary widely across isogenic cells in a population; (i


Original text

ABSTRACT In cellular regulatory networks, genetic ac- tivity is controlled by molecular signals that determine when and how often a given gene is transcribed. In genetically controlled pathways, the protein product encoded by one gene often regulates expression of other genes. The time delay, after activation of the first promoter, to reach an effective level to control the next promoter depends on the rate of protein accumulation. We have analyzed the chemical reactions con- trolling transcript initiation and translation termination in a single such ‘‘genetically coupled’’ link as a precursor to modeling networks constructed from many such links. Sim- ulation of the processes of gene expression shows that proteins are produced from an activated promoter in short bursts of variable numbers of proteins that occur at random time intervals. As a result, there can be large differences in the time between successive events in regulatory cascades across a cell population. In addition, the random pattern of expression of competitive effectors can produce probabilistic outcomes in switching mechanisms that select between alternative regu- latory paths. The result can be a partitioning of the cell population into different phenotypes as the cells follow dif- ferent paths. There are numerous unexplained examples of phenotypic variations in isogenic populations of both pro- karyotic and eukaryotic cells that may be the result of these stochastic gene expression mechanisms.
In all organisms, networks of coupled biochemical reactions and feedback signals organize developmental pathways, me- tabolism, and progression through the cell cycle. For example, overall coordination of the cell cycle results from an overar- ching set of dependent pathways in which the initiation of late events is dependent on the earlier events and the whole operates as a form of biochemical machine. Within these regulatory networks, genetic activity is controlled by molecular signals that determine when and how often a given gene is transcribed. Additional signals stimulated by environmental influences or by signals from other cells can affect the ongoing reactions to influence the future course of cellular events. Since a regulatory protein may act in combination with other signals to control many other genes, complex branching net- works of interactions are possible. In these nets, one regulatory protein can control genes that produce other regulators, that in turn control still other genes.
How long does it take for these messages and controlling influences to move through a regulatory cascade? In biochem- ical regulatory networks, the time intervals between successive events are determined by the inevitable delays while signal molecule concentrations either accumulate or decline. Genet- ically coupled links are links where the protein product en-
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Copyright 􏰁 1997 by THE NATIONAL ACADEMY OF SCIENCES OF THE USA 0027-8424􏰀97􏰀94814-6$2.00􏰀0
PNAS is available online at http:􏰀􏰀www.pnas.org.
coded by one gene regulates expression of other genes. The time delay in genetically coupled links (Fig. 1) depends on the time required for protein concentration growth, after pro- moter activation, to the concentration range that controls the next level in the cascade. Conversely, the time delay after the controlling promoter turns off depends on the time for the protein concentration to decay below the effective range. Fig. 1B shows a common architecture for such genetically coupled links. In these links, for appropriate combinations of input signals, transcripts are initiated and the protein product ac- cumulates when production exceeds degradation; the increas- ing protein concentration simply broadcasts the information that the promoter is ‘‘on.’’ The message is ‘‘received’’ or detected by the concentration-dependent response at the protein signal’s site(s) of action, stimulating a response at each site in accord with that site’s chemical behavior. (We use the term ‘‘protein signal’’ to mean the regulatory protein concen- tration in its effective form at its site of action.)
In this paper we examine the properties of a single geneti- cally coupled link as a precursor to modeling networks con- structed from many such links. Specifically, we ask what determines the time required for protein concentration to grow to effective signaling levels after a promoter is activated and how statistical variations in this time can affect observed cellular phenomena across a cell population. It has been proposed that the pattern of protein concentration growth is stochastic, exhibiting short bursts of variable numbers of proteins at varying time intervals (2, 3). (Herein the term ‘‘stochastic’’ is used in the statistical sense of resulting from a random process.) We formalize and quantify this notion of randomness in genetic regulatory mechanisms by explicitly characterizing the statistics of the random processes implicit in the chemical reactions (4). By analogy to electrical circuits, we will refer to this time interval between the switching on of the first promoter and activation or repression of the second promoter as a ‘‘switching delay.’’ There is also a switching delay of a different magnitude for the inverse functions when the controlling promoter is switched off. We are neglecting here the case where multiple molecules act combinatorially to determine the controlling action.
Then, as a concrete illustration of switching delays over a genetically coupled link, we simulate a representative link using parameters characteristic of links in bacterial regulatory networks. The simulation results show that short-term fluctu- ations in protein production can be large relative to signal thresholds that control expression of critical genes. For the same link in different cells of the same genotype, there will be wide random variations in both the times to produce a given protein concentration or in the number of proteins produced when the promoter is transiently activated. Implications of this noisy pattern of gene expression for cellular regulation include: (i) the switching delay for genetically coupled links, hence the time for the cell to execute cascaded functions, can vary widely across isogenic cells in a population; (i

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