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This has profound implications for the modelling of intracellular biochemical networks in which some of the reactions occur via an effective lower-dimensional motion.
Основные научные достижения
The use of combinations of three-dimensional in the cytosol and one or two-dimensional diffusion e. This facilitates target location [ 55 ] and hence it is likely quite common inside cells, e. Next we briefly discuss the predictions of stochastic and discrete models with those of the corresponding deterministic and continuous models for a given biological or biochemical system. Indeed there are many cases in which a deterministic description suffices to capture the approximate, qualitative physiological behaviour exhibited by a real biological system.
Some examples include models of the budding yeast cell cycle [ 58 ], of the glycolysis pathway in yeast [ 59 ] and L. However, there are also many instances in which it has been shown that stochastic effects lead to physiological predictions which cannot be reproduced qualitatively or quantitatively by the corresponding deterministic models. Prominent examples of such cases occur in models of biochemical oscillations e. In some cases noise may even induce oscillations in systems in which the corresponding deterministic model predicts that no oscillations are possible over the entire parameter space.
For example, noise may induce cycles in reaction schemes involving just two chemical agents whereas deterministic models predict that at least three chemical agents are necessary [ 63 ]. In general, it has been found that fluctuations may lead to steady-states in chemical reaction-diffusion systems which are not predicted by RDEs [ 6 ]. The parameter V x-axis is the gene transcription rate whereas M p frequency y-axis is the frequency of the mRNA concentration cycles.
The colour represents the log power of the frequency. Note that oscillations are paradoxically enhanced by noise; for low transcription rates, the stochastic models predict significant oscillations whereas the deterministic one predicts none. Reprinted from Journal of Theoretical Biology, Vol. Hamid, Control of internal and external noise in genetic regulatory networks, pp. In this essay we have discussed the four main methodologies for modelling reaction kinetics in cells Figure 3.
The determination of which model is most relevant for the analysis of a particular intracellular reaction pathway is closely tied to the physical nature of the intracellular region in which it occurs and also with the typical chemical concentrations involved in the reaction. Schematic showing the four modelling strategies and the conditions underlying their use and validity. The magnitude of the Kuramoto lengths and the intermolecular distances compared to the system size determines whether the homogeneity and continuum assumptions hold, respectively. In particular one needs to estimate the Kuramoto length scale, the size of the intracellular region in which reaction occurs and the average inter-molecular distances.
These can be estimated from knowledge of the diffusion coefficients, the reaction constants and molecular concentrations. Roughly speaking, reactions involving concentrations in the millimolar range will probably satisfy the continuum assumption whereas those involving smaller concentrations, such as nanomolar ones, will not.
Reactions occurring exclusively in a small intracellular compartment will favour the homogeneity assumption; those involving multiple transport between several compartments and the cytosol or in regions where molecular crowding is significant will not satisfy this assumption. As mentioned in the introduction, choosing the right model for the application at hand ensures the appropriate interpretation of experimental data. An example of this is the analysis of data obtained from single-molecule imaging techniques SMI [ 64 - 68 ].
The large fluctuations inherent in this data would necessarily have to be ignored if analysis proceeds via a deterministic framework. However, analysis via the stochastic framework of master equations has shown that such fluctuations may contain important information about molecular behaviour; for example, it was recently shown that fluctuations in the rate constants of a single enzyme yield information regarding its conformational dynamics [ 33 , 69 ].
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We conclude by noting that in general it is not possible to a priori rule out the effects of noise or space on a given intracellular reaction scheme of interest. It is important to be aware of the fact that non-spatial deterministic models based on classical REs constitute only a first approach to modelling intracellular dynamics; their physiological predictions should be carefully checked against those of spatial stochastic models which correspond more closely to reality.
Noise may induce novel and qualitatively different physiological behaviour than that predicted by deterministic models. We are very grateful to Michelle L.
Wynn for her critical comments during the revision of this essay. Any opinions, findings, and conclusions or recommendations expressed in this material do not necessarily reflect the views of the NIH. National Center for Biotechnology Information , U. Author manuscript; available in PMC Sep 3. Find articles by Santiago Schnell. The publisher's final edited version of this article is available at Essays Biochem. See other articles in PMC that cite the published article. Abstract In the past decade, advances in molecular biology such as the development of non-invasive single molecule imaging techniques have given us a window into the intricate biochemical activities that occur inside cells.
Introduction Over the past decade there has been an explosion of interest in methods for studying intracellular reactions, in particular those focusing on the determination of kinetic parameters [ 1 ]. Classical models of chemical reaction kinetics All reaction pathways can be decomposed into a series of elementary reaction steps, each of which is either unimolecular or bimolecular. The rate equation RE for species C is then given by the familiar classical equation: Can we understand intracellular reactions using classical models?
Discrete and stochastic models of chemical reaction kinetics Discrete models i. Applying the laws of probability yields the following equation describing the time evolution of the system: Stochastic kinetics in heterogeneous conditions We now consider the case where the molecular Kuramoto lengths are much smaller than the container thus leading to significant spatial heterogeneity in the concentrations. Deterministic models As remarked in the previous section, the majority of current stochastic models of intracellular kinetics assume well-mixing in the intracellular region where the reactions occur.
Open in a separate window. Discussion In this essay we have discussed the four main methodologies for modelling reaction kinetics in cells Figure 3. Acknowledgments We are very grateful to Michelle L. Schnell S, Turner TE. Reaction kinetics in intracellular environments with macromolecular crowding: Schnell S, Maini PK. A century of enzyme kinetics: Reliability of the K M and v max estimates.
Comments on Theoretical Biology. Mathematical and computational techniques to deduce complex biochemical reaction mechanisms. Stochastic processes in physics and chemistry. Togashi Y, Kaneko K. Molecular discreteness in reaction-diffusion systems yields steady states not seen in the continuum limit. Ben-Avraham D, Havlin S. Diffusion and reaction in fractals and disordered systems. Cambridge University Press; Cambridge: Single-molecule analysis of chemotactic signaling in Dictyostelium cells. Single-molecule analysis of epidermal growth factor signaling that leads to ultrasensitive calcium response.
Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. Journal of Chemical Physics. Absolute quantification of proteins and phosphoproteins from cell lysates by tandem MS. Stochastic simulation of the mammalian circadian clock. Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli.
It's a noisy business! Genetic regulation at the nanomolar scale. Stochastic gene expression in a single cell. Trafficking of a ligand-receptor complex on the growth cones as an essential step for the uptake of nerve growth factor at the distal end of the axon: Garland Science; New York: Molecular Biology of The Cell. In vivo nuclear magnetic resonance studies of glycolytic kinetics in Lactococcus lactis. Properties and metabolism of the aqueous cytoplasm and its boundaries. American Journal of Physiology. Pawson T, Scott JD. Signaling through scaffold, anchoring, and adaptor proteins.
A funny thing happened on the way to the Krebs cycle. News in Physiological Sciences. Cytoarchitecture and physical properties of cytoplasm: A rigorous derivation of the chemical master equation. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry. Stochastic simulation of chemical kinetics.
Annual Review of Physical Chemistry. Stochastic approaches for modelling in vivo reactions. Computational Biology and Chemistry. Stochastic simulation of large-scale genetic regulatory networks. Journal of Bioinformatics and Computational Biology. Modeling biological systems using Dynetica - a simulator of dynamic networks.
Growth and proliferation of mitochondria in embriogenesis of the sea urchin Paracentrotus lividus. Biologicheskie Membrany , 17 5: Biochemical and Biophysical Research Communications , 2: Biologicheskie Membrany , 17 2: Biologicheskie Membrany , 17 1: Contributions of plus and minus end pathways to microtubule turnover. Biology of the Cell , 91 6: Biologicheskie Membrany , 16 1: Molecular Biology of the Cell , 9: Biologicheskie Membrany , 15 6: Bioscience Reports , 18 4: Biologicheskie Membrany , 15 4: Molecular Biology of the Cell , 8: FEBS Letters , 1: Biologicheskie Membrany , 14 2: Biologicheskie Membrany , 14 1: Molecular Biology of the Cell , 7: A computerized reconstruction analysis.
Cell Biology International , 20 Biologicheskie Membrany , 13 5: High voltage electron microscopic study. Molecular Biology , Biofizika , 41 1: Molecular Biology of the Cell , 6: Journal of Photochemistry and Photobiology B-Biology , 29 2: Journal of Submicroscopic Cytology and Pathology , 27 3: Cell Biology International , 19 2: Journal of Structural Biology , 3: Genetika , 30 3: Rebecca Heald - Publications Affiliations: Mitotic spindle assembly and function in Xenopus laevis. A versatile multivariate image analysis pipeline reveals features of Xenopus extract spindles.
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Thirty years of search and capture: The complex simplicity of mitotic spindle assembly. Biological Scaling Problems and Solutions in Amphibians.
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Cold Spring Harbor Perspectives in Biology. Glutamylation of Nap1 modulates histone H1 dynamics and chromosome condensation in Xenopus. RUVs drive chromosome decondensation after mitosis. TPX2 levels modulate meiotic spindle size and architecture in Xenopus egg extracts. Morphology and function of membrane-bound organelles. Current Opinion in Cell Biology. Cytoplasmic volume modulates spindle size during embryogenesis.
Modelling reaction kinetics inside cells
Science New York, N. Interplay between spindle architecture and function. International Review of Cell and Molecular Biology. Molecular Biology of the Cell. N-terminal phosphorylation of p60 katanin directly regulates microtubule severing. Journal of Molecular Biology. Histone H1 compacts DNA under force and during chromatin assembly.
Mechanisms of intracellular scaling. Annual Review of Cell and Developmental Biology. Altering membrane topology with Sar1 does not impair spindle assembly in Xenopus egg extracts. Multiple domains of human CLASP contribute to microtubule dynamics and organization in vitro and in Xenopus egg extracts. Mitotic spindle assembly around RCC1-coated beads in Xenopus egg extracts. Katanin contributes to interspecies spindle length scaling in Xenopus.
Mitotic chromosome size scaling in Xenopus. Cell Cycle Georgetown, Tex. Atomic force microscope imaging of chromatin assembled in Xenopus laevis egg extract. Cell structure and dynamics. A computational model predicts Xenopus meiotic spindle organization.
Xenopus egg extracts increase dynamics of histone H1 on sperm chromatin. Centrosome proteins form an insoluble perinuclear matrix during muscle cell differentiation. Mitotic spindle assembly mechanisms The Kinetochore: From Molecular Discoveries to Cancer Therapy. Interview by Caitlin Sedwick. Journal of Cell Science. Discovery of selective aminothiazole aurora kinase inhibitors. Mechanisms of mitotic spindle assembly and function. International Review of Cytology.