Limits...
Biochemical fluctuations, optimisation and the linear noise approximation.

Pahle J, Challenger JD, Mendes P, McKane AJ - BMC Syst Biol (2012)

Bottom Line: Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species?We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK.We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

View Article: PubMed Central - HTML - PubMed

Affiliation: School of Computer Science and Manchester Centre for Integrative Systems Biology, The University of Manchester, 131 Princess Street, Manchester, UK. juergen.pahle@manchester.ac.uk

ABSTRACT

Background: Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. However, the systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms. This is particularly problematic when, as is often the case, some or many model parameters are not well known. Here, we propose a solution to this problem, namely a combination of the linear noise approximation with optimisation methods. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system. Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Examples are, what is the lowest amplitude of stochastic fluctuations possible within given parameter ranges? Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species? Unlike stochastic simulation methods, this has no requirement for small numbers of molecules and thus can be applied to cases where stochastic simulation is prohibitive.

Results: We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK. Using our method we were able to quickly find local maxima of covariances between particle numbers in the ERK model depending on the activities of phospho-MKKK and its corresponding phosphatase. With the p38 MAPK model our method was able to efficiently find conditions under which the coefficient of variation of the output of the signalling system, namely the particle number of Hsp27, could be minimised. We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

Conclusions: Our strategy is a practical method for the efficient investigation of fluctuations in biochemical models even when some or many of the model parameters have not yet been fully characterised.

Show MeSH

Related in: MedlinePlus

Two-dimensional parameter scan of MKKK and MKK-P particle numbers' covariance in the ERK MAP model. A two-dimensional parameter scan of the covariance of the particle numbers of species MKKK and MKK-P. The parameter v2 was varied between 0.22 and 0.41 and the parameter k4 between 0.015 and 0.035.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3814289&req=5

Figure 4: Two-dimensional parameter scan of MKKK and MKK-P particle numbers' covariance in the ERK MAP model. A two-dimensional parameter scan of the covariance of the particle numbers of species MKKK and MKK-P. The parameter v2 was varied between 0.22 and 0.41 and the parameter k4 between 0.015 and 0.035.

Mentions: We then wanted to investigate the conditions under which fluctuations in chemical species at different positions of the signalling cascade become correlated. To achieve this, we used the optimisation task in COPASI to maximise the covariance of the fluctuations of MKKK and MKK-P, allowing the reaction parameters v2 and k4 to vary over a given range of values. Using the evolutionary programming algorithm [28] (which took 199 seconds to run) 4004 steady state and LNA evaluations were carried out. A local maximum of the covariance was found with a value of 4035 particles2 for v2 = 0.3226 and k4 = 0.0166. The algorithm converged to this value already after 880 iterations. A parameter scan over the same parameter space was also performed to better illustrate the change in correlation with these two parameters. Figure 4 shows how the covariance of the fluctuations of MKKK and MKK-P varies with the reaction parameters, and provides a visualisation of the landscape that the optimisation algorithm must traverse. Note that the covariance becomes negative for some parameter values.


Biochemical fluctuations, optimisation and the linear noise approximation.

Pahle J, Challenger JD, Mendes P, McKane AJ - BMC Syst Biol (2012)

Two-dimensional parameter scan of MKKK and MKK-P particle numbers' covariance in the ERK MAP model. A two-dimensional parameter scan of the covariance of the particle numbers of species MKKK and MKK-P. The parameter v2 was varied between 0.22 and 0.41 and the parameter k4 between 0.015 and 0.035.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3814289&req=5

Figure 4: Two-dimensional parameter scan of MKKK and MKK-P particle numbers' covariance in the ERK MAP model. A two-dimensional parameter scan of the covariance of the particle numbers of species MKKK and MKK-P. The parameter v2 was varied between 0.22 and 0.41 and the parameter k4 between 0.015 and 0.035.
Mentions: We then wanted to investigate the conditions under which fluctuations in chemical species at different positions of the signalling cascade become correlated. To achieve this, we used the optimisation task in COPASI to maximise the covariance of the fluctuations of MKKK and MKK-P, allowing the reaction parameters v2 and k4 to vary over a given range of values. Using the evolutionary programming algorithm [28] (which took 199 seconds to run) 4004 steady state and LNA evaluations were carried out. A local maximum of the covariance was found with a value of 4035 particles2 for v2 = 0.3226 and k4 = 0.0166. The algorithm converged to this value already after 880 iterations. A parameter scan over the same parameter space was also performed to better illustrate the change in correlation with these two parameters. Figure 4 shows how the covariance of the fluctuations of MKKK and MKK-P varies with the reaction parameters, and provides a visualisation of the landscape that the optimisation algorithm must traverse. Note that the covariance becomes negative for some parameter values.

Bottom Line: Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species?We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK.We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

View Article: PubMed Central - HTML - PubMed

Affiliation: School of Computer Science and Manchester Centre for Integrative Systems Biology, The University of Manchester, 131 Princess Street, Manchester, UK. juergen.pahle@manchester.ac.uk

ABSTRACT

Background: Stochastic fluctuations in molecular numbers have been in many cases shown to be crucial for the understanding of biochemical systems. However, the systematic study of these fluctuations is severely hindered by the high computational demand of stochastic simulation algorithms. This is particularly problematic when, as is often the case, some or many model parameters are not well known. Here, we propose a solution to this problem, namely a combination of the linear noise approximation with optimisation methods. The linear noise approximation is used to efficiently estimate the covariances of particle numbers in the system. Combining it with optimisation methods in a closed-loop to find extrema of covariances within a possibly high-dimensional parameter space allows us to answer various questions. Examples are, what is the lowest amplitude of stochastic fluctuations possible within given parameter ranges? Or, which specific changes of parameter values lead to the increase of the correlation between certain chemical species? Unlike stochastic simulation methods, this has no requirement for small numbers of molecules and thus can be applied to cases where stochastic simulation is prohibitive.

Results: We implemented our strategy in the software COPASI and show its applicability on two different models of mitogen-activated kinases (MAPK) signalling -- one generic model of extracellular signal-regulated kinases (ERK) and one model of signalling via p38 MAPK. Using our method we were able to quickly find local maxima of covariances between particle numbers in the ERK model depending on the activities of phospho-MKKK and its corresponding phosphatase. With the p38 MAPK model our method was able to efficiently find conditions under which the coefficient of variation of the output of the signalling system, namely the particle number of Hsp27, could be minimised. We also investigated correlations between the two parallel signalling branches (MKK3 and MKK6) in this model.

Conclusions: Our strategy is a practical method for the efficient investigation of fluctuations in biochemical models even when some or many of the model parameters have not yet been fully characterised.

Show MeSH
Related in: MedlinePlus