Robustness in the Chemotaxis Network of Escherichia Coli - Coursework Example

Robustness in the Chemotaxis Network of Escherichia coli. INTRODUCTION A biochemical network is a group of interrelated processes whose coordinated action performs a particular function in an organism. If we consider the case of single celled organisms, their cellular biochemical networks work in close relation with each other in order to fulfil the conditions necessary for the smooth functioning of the cell. A sudden decrease or increase cellular molecular concentrations can cause the malfunction of several functions which are vital for the cell’s survival. Due to their interconnectedness biological networks are very unstable. Biological networks could achieve stability by one of the following ways; either the network functions such that the values of the various biochemical parameters it comprises are regulated in a very precise manner in order to achieve a particular function and hence stability, if these parameters do not have these precise values the network becomes unstable and malfunctions or the biochemical network functions such that its key properties are insensitive to variations in the precise values of biochemical parameters. This report focuses on this second possibility; we take a close look at robustness or insensitivity of the key properties of the biochemical network responsible for bacterial chemotaxis in Escherichia coli to changes in various biochemical parameters. The motion of bacteria like Escherichia coli is similar to a random walk problem, the motion is a “smooth run” interspersed by brief tumbling periods during which the bacterium chooses a new direction. The bacterium moves away from repellents or towards attractants by modifying its tumbling frequency. THE CHEMOTAXIS SYSTEM OF Escherichia coli Chemotaxis is the directed movement of a cell or organism towards or away from a chemical source. There are many proteins involved in chemotactic response. Following Alon & al1 we look at the main features of the chemotaxis network of Escherichia coli. The motion of an Escherichia coli bacterium can be likened to a random walk; it is a smooth swimming motion which is interrupted by brief periods of tumbling motion during which the bacterium randomly chooses a new direction. The bacterium achieves chemotaxis by changing its tumbling frequency. The process of changing the direction of its motion by varying the tumbling frequency is controlled by a protein network. Chemoreceptors span the cellular membrane of the bacterium and are responsible for sensing the external environment of the bacterium and are also sensitive to external chemical stimuli. Chemotactic ligands which stimulate the bacterium become attached to specialised chemoreceptors (MCP), the MCP forms stable complexes (E) with the kinases Che A anb Che W. The kinase Che A adds phosphoryl groups to the response regulator, Che Y; the phosphorylated form of Che Y (Che Yp) then binds to the flagellar motors of the bacterium where it causes tumbling. The tumbling frequency can be modified by altering the kinase activity of Che A, this occurs when the chemotactic ligand binds with the chemoreceptor. The chemoreceptor also has the ability to undergo reversible methylation; this either enhances (in this case we have a methylation reaction) the kinase activity and adaptation to changes in chemotactic ligand concentration or impedes (in this case we have a demethylation reaction) it. Two proteins, Che R and Che B make adaptation possible. Che R catalyses the methylation of the chemoreceptor, while Che B catalyses its demethylation. A feed-back mechanism is achieved when Che A enhances the phosphorylation of Che B; this has the effect of increasing the demethylating function of Che B. MECHANISM FOR ROBUST ADAPTATION IN Escherichia coli The complex MCP + Che A + Che W which we call E can be modified; this modification reaction is catalysed by the enzyme R. We denote the modified form of the complex E by Em. The enzyme B catalyses the reverse modification reaction, that is the modification of Em to E. E (Em) has a probability Λ (Λm) which depends on the concentration of the chemotactic ligand which is the input of this system. Robust adaptation occurs when R works at saturation and B acts only on the active form of Em. THE TWO STATE MODEL In what follows we present an analysis of the chemotactic network of Escherichia coli; we use a simple two-state model to achieve this end. This model assumes that the receptor complex can occupy two states namely the active and inactive states. While in the active state the receptor complex shows kinase activity and it phosphorylates the response regulator complex; the phosphoprylated response regulator complex then induces tumbling by sending a tumbling signal to the flagellar motors. The probability that the receptor complex is in the active or inactive state is determined by its methylation level and ligand occupancy. The two-state model faithfully reproduces the main features of biological chemotaxis: when a model biological system is subjected to a change in attractant or repellent concentration, it is able to respond and adapt to the change imposed. A thorough description of the model consists of a set of coupled differential equations which describe the interactions between various protein components which make up the network. The receptor complex MCP + Che A + Che W which is considered here as a single entity E is the main component of the two state model. The complex has two states namely the active and inactive states. In the active state the receptor complex shows kinase activity of Che A, this achieved by phosphorylation of the response regulator, Che Y and the phosphorylated response regulator then induces tumbling by sending a tumbling signal to the flagellar motors. The system activity is denoted as A, it is given by the average number of receptors in the active state. The system activity is a measure of the tumbling frequency of the bacteria. The treatment could be made more rigorous by considering the fact that the transformation of A into the tumbling frequency is dependent on ; (i) the rate of Che Y phosphorylation and dephosophorylation and (ii) the interaction of Che Y with the flagellar motors. We consider a complex with m methylation sites, where m is an integer between 1 and M inclusive, M is the Mth methylation site. These complexes can either be occupied or unoccupied by chemotactic ligands. We denote a methylated complex which is occupied (unoccupied) by a chemotactic ligand by Eom (Eum). Also each occupied (unoccupied) receptor has a probability Λom (Λm) of being in the active state. Also l, B, R, EumB represent the ligand concentration, Che B concentration, Che R concentration and the EumChe B complex respectively and so on. Following Barkai and Leibler3, the differential equations governing the model can be written as: dEum / dt = -Kl Eum + K-lEom + (1 – δm, o) [-ab Λm EumB + db EumB + KrEum+1R] + (1 – δm, M) [-ar ΛmEumR + a’r (1- Λm) EumR + dbEumR + KbEum+1B] (1) (m = 1 … M) The presence of Λm in the equation above is due to the fact that Che B demethylates only the active receptors, the term ar (a’r) above represents the association rate constant of Che R to the active (inactive) receptors. δa, b is the Kronecker’s delta (it satisfies the following; δa, b = 1, when a = b and δa, b = 0, otherwise). The variations of other complexes present in the system for example EumR or EumB can be characterised in a manner similar to that of the equation above. If Λom and Λm are kept constant then the biochemical parameters of the system allow for nine rate constants (Kl, K-l, ar, a’r, dr, Kr, ab, db, Kb) and three enzyme concentrations, that is the total Che R, Che B and receptor complex concentrations. Rather than embark on a long and tedious mathematical treatment we focus on principles and biochemical interpretations or inferences. Accordingly the following points are worthy of mention here. Firstly the model presented here is by no means the only one of its kind which faithfully reproduces the main features of a robust chemotactic network. Instead it is one of the simplest forms of the two-state model which is in excellent agreement with experimental values for adaptation and response in the wild strain of Escherichia coli. Furthermore the main assumptions made in our analysis are outlined below; (i) Since the ligand concentration is the main input to the system, the rate of binding and unbinding of the ligand to the receptor affects the activity of the complex, hence the assumption that the “binding affinity” of the ligand is independent of receptor activity and the extent of receptor methylation. (ii) The methylation and demethylation reactions occur over relatively long time-scales and all active receptors are demethylated at the same rate. The phosphorylation of Che B is assumed to be irrelevant, that is the phosphorylated Che B molecules can not move freely in the cell; instead the phosphorylated Che B is only able to demethylate the same receptor that phosphorylated it. If Che B and its phosphorylated form demethylate only active receptors then the system is able to adapt to any change in the values of its biochemical parameters rapidly. (iii) The methylating enzyme Che R acts both on active and inactive receptors. RESULTS The most important finding of this study is the fact that the model shows a near perfect adaptation in situations characterised by wide-scale variations in the values of the biochemical parameters of the chemotactic network. In what follows the main points of our findings are outlined. First, the version of the two-state model used here assumes that the enzyme kinetics is in a quasi-steady-state, in other words Michaelis-menten kinetics is assumed. Secondly, the model has the ability to quickly adapt to changes in enzymatic reaction rates though and the adaptation property also behaves similarly in response to changes in the number of methylation sites in a range of different systems. Thirdly, it is realised that only some properties of a chemotactic network are insensitive to changes in the values of the biochemical parameters of the network. One such property is the adaptation precision of the network. This can be seen from the way in which the adaptation precision (P) and the adaptation time (τ) respond to a step-like variation in the saturation amount of the attractant expressed as a function of the total parameter variation (K).When we compare an altered model system, which can be obtained by decreasing or increasing the values of all biochemical parameters by a factor of two in an arbitrary fashion and a reference system, we discover that the P of the altered model system has a probability of 0.95 as opposed to 1.00 for the reference system and that the probability that the τ of the altered model system deviates from that of the reference system is less than 5 %. Also the inverse steady-state activity (A-1) of the altered model system varies proportionately with the τ, this is in good agreement with the same variation for the reference system. As such we can conclude that only specific network properties and not the chemotactic network are insensitive to variations in the values of the various biochemical parameters of the network. DISCUSSION A central question still remains unanswered namely “what features of the chemotactic network make the adaptation property robust”. In what follows the main features of a simple mechanism for “robust adaptation” are outlined. This mechanism is based on the bacterial chemotactic network of Escherichia coli. Following the treatment of Barkai and Leibler2, assume for simplicity an enzyme (E) which is sensitive to external signals such as a chemotactic ligand (l). Each enzyme molecule is at equilibrium between an active state in which it catalyses a reaction and an inactive state in which it does not perform any catalytic function. The chemotactic ligand concentration affects the equilibrium between the active and inactive states of the enzyme; let us assume that a change in the chemotactic ligand concentration induces a rapid response of the system which shows up as a shift in the equilibrium of E. As such the chemotactic ligand concentration is the input of this signal interpretation system while the active enzyme concentration which is directly proportional to the system activity (A) is the output of the system. The enzyme E can be modified in a reversible manner, this modification can be achieved either by methylation or phosphorylation. This modification of the enzyme E has repercussions on the probability of it being in the active or inactive state and could reduce the effect of the chemotactic ligand concentration on the system. Summarily the system activity expressed as a function of the chemotactic ligand concentration is given by; A(l) = Λ(l)E + Λm(l)Em (2) Where Λ (Λm) is the probability that the unmodified (modified) enzyme is active, while E and Em are the concentrations of the unmodified and modified enzyme respectively. Initially the system responds rapidly to a change in the input level of the chemotactic ligand concentration, but eventually on large time-scales the system activity evolves following quasi-steady-state or Michaelis-Menten enzyme kinetics. When the steady-state activity (Ast) is independent of the chemotactic ligand, then the system is said to be “adaptive”. Two mechanisms can be envisaged in order to achieve this end; either there is a “fine-tuned” dependence of biochemical parameters on the level of the chemotactic ligand concentration which results in the system being adaptive or the rate of the modification and reverse modification reactions only depend on the system activity (active enzyme concentration) and not on the concentration of the modified and unmodified enzymes. The latter case is an example of “activity-dependent kinetics”, here the steady-state activity is independent of the chemotactic ligand concentration and hence the system is adaptive. The system can be looked upon as a feed-back system, which functions such that the output which is the active enzyme concentration or system activity determines the rate of the modification reaction which then determines the variation of the system activity. Let us now consider a specific system as an example. Following Barkai and Leibler2 once more we consider a system for which only the modified enzyme is active ( that is Λ = 0 ), in this case the enzyme R which catalyses the modification reaction E→Em, works at saturation and the enzyme B which catalyses the reaction Em→E, can only bind to active enzymes. The result is a constant modification rate, while on the other hand the reverse modification rate is then a simple function of the system activity or active enzyme concentration, this can be expressed mathematically as; dEm / dt = VRmax − VBmax A (3) Kb + A In equation (3) above, VRmax and VBmax are the maximal velocities of the modification and reverse modification reactions respectively, while Kb is the Michaelis constant for the reverse modification reaction. It is assumed here that the condition VRmax < VBmax holds and that the enzyme kinetics follows quasi-steady-state or Michaelis-Menten kinetics. The functioning of this feed-back system can be analysed as follows. The system activity (A) is repeatedly compared to a reference steady-state value (Ast) which is given by the expression below; Ast = Kb VRmax (4) VBmax − VRmax Case one (A < Ast) In this case the rate of modification of the enzyme increases and this results in an increase in A. Case two (Ast > A) In this case the rate of modification reduces and this results in a reduction in A. As such the system shows adaptive robustness, since the system always returns to its steady-state system activity. The activity-dependent kinetics results in the adaptation property being insensitive to variations in the values of the biochemical parameters of the system. Furthermore the enzyme concentration determines the steady-state activity and as such the latter is not a robust property of the chemotactic network. The system activity can thus be monitored over relatively large time-scales; this monitoring function can be achieved by modifying the enzyme concentration while keeping the adaptation property constant over shorter time-scales. There are two observations which reinforce the plausibility of the existence of a robust adaptation mechanism for bacterial chemotaxis, rather than a fine-tuned version of the mechanism; Firstly, observation of various chemotactic bacterial populations reveals the robust nature of the adaptation property. Indeed a robust mechanism for bacterial chemotaxis possesses features which allow for wide-scale variations of certain characteristics in bacterial populations. The only mechanism which allows for variations in the values of the biochemical parameters of the chemotactic network, while preserving the adaptation property or adaptive robustness is the robust mechanism of chemotactic networks which is the subject of the present study. Accordingly chemotactic bacteria can accommodate changes in the values of the biochemical parameters of their chemotactic networks; this explains for example why chemotactic bacteria can accommodate genetic polymorphism even though this usually leads to a variation of the biochemical parameters of the chemotactic network. Secondly, some features of a bacterium’s chemotactic response like the values of the steady-state tumbling frequency and the adaptation time vary from one individual to the other in a population of genetically identical chemotactic bacteria. On the contrary the adaptation property is preserved for the whole population of genetically identical chemotactic bacteria. This occurrence can be accounted for by the model for robust adaptation presented here. Let us look at a specific example; the concentration of the methylating enzyme Che R, which is a cellular protein, is very low and as such this concentration is usually subject to “stochastic variations”, consequently the adaptation time and the steady-state tumbling frequency which are not robust properties of the chemotactic network vary widely. In addition, the model presented here predicts that these variations would be of a strongly correlative nature; interestingly enough this has already been observed experimentally. The results obtained here allow us to do much more than just explaining how the response and adaptation of the chemotactic network come about. The model considered in this study also works well in explaining similar or related behaviour in different bacteria which exhibit the same network. Speaking more concretely, system activity uniquely determines network dynamics, accordingly the system must respond and adapt to any change which affects its system activity. In addition, the findings of this study have important wide scale implications; contrarily to a fine-tuned mechanism for adaptation, the mechanism for robust adaptation as presented here could be applied to other so called “signal transduction networks”. This is so because robustness could be a very important and common feature of many systems, and it could be a factor which is crucial to cellular survival. This possibility makes sense since all organisms are constantly subjected to a myriad of external stimuli and they must adapt to changes in their external environment. As such systems whose main properties are robust would have a competitive edge over other systems, for example such systems would adapt more easily to environmental changes. Finally, the complexity inherent to most biological systems limits the extent of any analysis which tries to discover how they function. The greatest limitation of such studies is the difficulty of isolating smaller subsystems that can be studied separately. For example the model presented here is limited. Firstly, it does not take into account the differences which exist between different varieties of receptors and their chemical interactions. In addition, the effect of other cellular processes and cell components which are external to the chemotactic network are neglected. Furthermore, the complexity of biological systems is such that all attempts to understand their complete molecular description are futile. Usually reaction rates and enzyme concentrations can only be measured in vitro, these values may be different from those prevailing in vivo and consequently the conclusions which are inferred from them may be deeply flawed. Robustness may be the only way out of such a complicated situation. Since robust properties are insensitive to changes in the values of the biochemical parameters of the network, they would also most probably be insensitive to changes induced by external networks. Thus in principle it should be possible to for us understand at least some aspects of cell functioning without a full knowledge of cellular biochemistry. METHODS Numerical integration of the kinetics equations defining the two-state model was used to investigate its properties. Computer programs in C++ language were executed on an SGI (R4000) workstation using a standard routine. Typical CPU time for finding a numerical solution of a model system is of order 1 min. A particular model system was obtained by assigning values to the rate constants and the total enzyme concentrations. Most of our results were obtained for a reference system defined by the following biochemical parameters: the equilibrium binding constant of ligand to receptor is 1μM and the time constant for the reaction is 1ms. Che R methylates both active and inactive receptors at the same rate, with a Michaelis constant of 1.25 μM, a time constant of 10s, Che B (Che Bp) demethylates only active receptors with a Michaelis constant of 1.25 μM and a time constant of 10s. The numbers of enzyme molecules per cell are: 10,000 receptor complexes, 2,000 Che B and 200 Che R (cell volume = 1.4 x 10-15l). The probabilities that a receptor with m = 1,…4 methylation sites is in its active state are: Λ1 = 0.1, Λ2 = 0.5, Λ3 = 0.75, Λ4 = 1 if it is occupied, and Λ01 = 0, Λ02 = 0.1, Λ03 = 0.5, Λ04 = 1 if it is occupied. REFERENCES 1. Alon, U., Surette, M.G., Barkai, N. & Leibler, S. Robustness in bacterial chemotaxis. Nature. 397, 168 – 171 (1999). 2. Barkai, N. and Leibler, S. 1997. Robustness in simple biochemical networks. Nature. 387, 913 – 917 (1997). 3. Hazelbauer, G. L., Berg, H. C. & Matsumura, P. M. Bacterial motility and signal transduction. Cell 73, 15–22 (1993). 4. Bourret, R. B., Borkovich, K. A. & Simon, M. I. Signal transduction pathways involving protein phosphorylation in prokaryotes. Annu. Rev. Biochem. 60, 401–441 (1991). 5. Adler, J. Chemotaxis in bacteria. Annu. Rev. Biochem. 44, 341–356 (1975). 6. Bray, D., Bourret, R. B. & Simon, M. I. Computer simulation of the phosphorylation cascade controlling bacterial chemotaxis. Mol. Biol. Cell 4, 469–482 (1993). 7. Adler, J. Chemotaxis in bacteria. Science 153, 708–716 (1996). 8. Berg, H. C. & Brown,D. A. Chemotaxis in E. coli analysed by three-dimensional tracking. Nature 239, 500–504 (1972). 9. Macnab, R. M. & Koshland, D. E. The gradient-sensing mechanism in bacterial chemotaxis. Proc. Natl Acad. Sci. USA 69, 2509–2512 (1972). 10. Block, S. M., Segall, J. E. & Berg, H. C. Impulse responses in bacterial chemotaxis. Cell 31, 215–226 (1982). 11. Koshland, D. E. A response regulator model in a simple sensory system. Science 196, 1055 (1977). 12. Berg, H. C. & Tedesco, P. M. Transient response to chemotactic stimuli in E. coli. Proc. Natl Acad. Sci. USA 72, 3235–3239 (1975). 13. Springer, M. S., Goy, M. F. & Adler, J. Protein methylation in behavioural control mechanism and in signal transduction. Nature 280, 279–284 (1979). 14. Koshland, D. E., Goldbeter, A. & Stock, J. B. Amplification and adaptation in regulatory and sensory systems. Science 217, 220–225 (1982). 15. Khan, S., Spudich, J. L., McCray, J. A. & Tentham, D. R. Chemotactic signal integration in bacteria. Proc. Natl Acad. Sci. USA 92, 9757–9761 (1995). 16. Segel, L. A., Goldbeter, A., Devreotes, P. N. & Knox, B. E. A mechanism for exact sensory adaptation based on receptor modification. J. Theor. Biol. 120, 151–179 (1986). 17. Hauri, D. C. & Ross, J. A model of excitation and adaptation in bacterial chemotaxis. Biophys. J. 68, 708–722 (1995). 18. Asakura, S. &Honda, H. Two-state model for bacterial chemoreceptor proteins. J. Mol. Biol. 176, 349– 367 (1984). 19. Spudich, J. L. & Koshland, D. E. Non-genetic individuality: chance in the single cell. Nature 262, 467– 471 (1976). 20. Kleene, S. J., Hobson, A. C. & Adler, J. Attractants and repellents influence methylation and demethylation of methyl-accepting proteins in an extract of E. coli. Proc. Natl Acad. Sci. USA 76, 6309–6313 (1979). Read More

Chemotactic ligands which stimulate the bacterium become attached to specialised chemoreceptors (MCP), the MCP forms stable complexes (E) with the kinases Che A anb Che W. The kinase Che A adds phosphoryl groups to the response regulator, Che Y; the phosphorylated form of Che Y (Che Yp) then binds to the flagellar motors of the bacterium where it causes tumbling. The tumbling frequency can be modified by altering the kinase activity of Che A, this occurs when the chemotactic ligand binds with the chemoreceptor.

The chemoreceptor also has the ability to undergo reversible methylation; this either enhances (in this case we have a methylation reaction) the kinase activity and adaptation to changes in chemotactic ligand concentration or impedes (in this case we have a demethylation reaction) it. Two proteins, Che R and Che B make adaptation possible. Che R catalyses the methylation of the chemoreceptor, while Che B catalyses its demethylation. A feed-back mechanism is achieved when Che A enhances the phosphorylation of Che B; this has the effect of increasing the demethylating function of Che B.

MECHANISM FOR ROBUST ADAPTATION IN Escherichia coli The complex MCP + Che A + Che W which we call E can be modified; this modification reaction is catalysed by the enzyme R. We denote the modified form of the complex E by Em. The enzyme B catalyses the reverse modification reaction, that is the modification of Em to E. E (Em) has a probability Λ (Λm) which depends on the concentration of the chemotactic ligand which is the input of this system. Robust adaptation occurs when R works at saturation and B acts only on the active form of Em.

THE TWO STATE MODEL In what follows we present an analysis of the chemotactic network of Escherichia coli; we use a simple two-state model to achieve this end. This model assumes that the receptor complex can occupy two states namely the active and inactive states. While in the active state the receptor complex shows kinase activity and it phosphorylates the response regulator complex; the phosphoprylated response regulator complex then induces tumbling by sending a tumbling signal to the flagellar motors.

The probability that the receptor complex is in the active or inactive state is determined by its methylation level and ligand occupancy. The two-state model faithfully reproduces the main features of biological chemotaxis: when a model biological system is subjected to a change in attractant or repellent concentration, it is able to respond and adapt to the change imposed. A thorough description of the model consists of a set of coupled differential equations which describe the interactions between various protein components which make up the network.

The receptor complex MCP + Che A + Che W which is considered here as a single entity E is the main component of the two state model. The complex has two states namely the active and inactive states. In the active state the receptor complex shows kinase activity of Che A, this achieved by phosphorylation of the response regulator, Che Y and the phosphorylated response regulator then induces tumbling by sending a tumbling signal to the flagellar motors. The system activity is denoted as A, it is given by the average number of receptors in the active state.

The system activity is a measure of the tumbling frequency of the bacteria. The treatment could be made more rigorous by considering the fact that the transformation of A into the tumbling frequency is dependent on ; (i) the rate of Che Y phosphorylation and dephosophorylation and (ii) the interaction of Che Y with the flagellar motors. We consider a complex with m methylation sites, where m is an integer between 1 and M inclusive, M is the Mth methylation site. These complexes can either be occupied or unoccupied by chemotactic ligands.

We denote a methylated complex which is occupied (unoccupied) by a chemotactic ligand by Eom (Eum). Also each occupied (unoccupied) receptor has a probability Λom (Λm) of being in the active state.

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