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1.
J Math Biol ; 83(5): 57, 2021 11 03.
Article in English | MEDLINE | ID: mdl-34731323

ABSTRACT

Inherent stochasticity in gene expression leads to distributions of mRNA copy numbers in a population of identical cells. These distributions are determined primarily by the multitude of states of a gene promoter, each driving transcription at a different rate. In an era where single-cell mRNA copy number data are more and more available, there is an increasing need for fast computations of mRNA distributions. In this paper, we present a method for computing separate distributions for each species of mRNA molecules, i.e. mRNAs that have been either partially or fully processed post-transcription. The method involves the integration over all possible realizations of promoter states, which we cast into a set of linear ordinary differential equations of dimension [Formula: see text], where M is the number of available promoter states and [Formula: see text] is the mRNA copy number of species j up to which one wishes to compute the probability distribution. This approach is superior to solving the Master equation (ME) directly in two ways: (a) the number of coupled differential equations in the ME approach is [Formula: see text], where [Formula: see text] is the cutoff for the probability of the jth species of mRNA; and (b) the ME must be solved up to the cutoffs [Formula: see text], which must be selected a priori. In our approach, the equation for the probability to observe n mRNAs of any species depends only on the the probability of observing [Formula: see text] mRNAs of that species, thus yielding a correct probability distribution up to an arbitrary n. To demonstrate the validity of our derivations, we compare our results with Gillespie simulations for ten randomly selected system parameters.


Subject(s)
RNA, Messenger , Probability , RNA, Messenger/genetics
2.
Phys Med ; 64: 33-39, 2019 Aug.
Article in English | MEDLINE | ID: mdl-31515033

ABSTRACT

Proton radiotherapy has a potential to provide an effective cancer treatment while sparing greater volume of healthy tissue than the conventional X-ray based radiotherapy. However, in lungs this potential is hindered by motion due to breathing. An important quantity in treatment verification is the correlation between the respiratory phases (RP) and the timing of pencil beam scanning (PBS). In this note, we demonstrate how the RP can be estimated using Prompt gamma (PG) detection profiles collected during a treatment. We utilized a 4D-CT of a patient with lung cancer, a treatment plan and a PG simulator. The treatment plan consisted of ten layers corresponding to ten proton energies. The RPs of the 4D-CT were interpolated using a deformable registration algorithm, so as to have fifty RPs in total. Deviations from regular breathing were introduced via time dependent frequency modulation. Fifty unique breathing patterns were generated, for which PG profiles were simulated for each pencil beam. Poisson noise was added to each PG profile to account for photon statistics. The RPs were estimated by comparing the PG profiles with and without Poisson noise via three different methods: the RP associated with each layer was estimated 1) independently of the other layers, 2) using a linear correlation between the layers, and 3) using a quadratic correlation between the layers. The best model, the quadratic model, yielded an average error in RP estimation relative to the breathing period of 5% of the breathing period or less with a 90% confidence interval.


Subject(s)
Four-Dimensional Computed Tomography , Gamma Rays , Proton Therapy/methods , Respiration , Humans , Lung Neoplasms/diagnostic imaging , Lung Neoplasms/physiopathology , Lung Neoplasms/radiotherapy , Monte Carlo Method , Radiotherapy Planning, Computer-Assisted , Time Factors
3.
J Math Biol ; 79(6-7): 2211-2236, 2019 12.
Article in English | MEDLINE | ID: mdl-31511967

ABSTRACT

Stochastic fluctuations in the copy number of gene products have perceivable effects on the functioning of gene regulatory networks (GRNs). The master equation (ME) provides a theoretical basis for studying such effects. However, solving the ME can be a task that ranges from simple to difficult to impossible using conventional methods. Therefore, discovering new techniques for solving the ME is an important part of research on stochastic GRNs. In this paper, we present a novel approach to obtaining the generating function (GF), which contains the same information as the ME, for a one gene system that includes multi-step post-transcription and post-translation processes. The novelty of the approach lies in solving the GF for proteins in terms of a particular path taken by the partially and fully processed mRNAs in the time-copy number plane, after which the GF is summed over all possible paths. We prove a theorem that shows the summation of all paths to be equivalent to an equation similar to the ME for the mRNAs. On a system with six gene products in total and randomly selected initial conditions, we confirm the validity of our results by comparing them with Gillespie simulations.


Subject(s)
Gene Regulatory Networks , Models, Genetic , Protein Processing, Post-Translational , RNA Processing, Post-Transcriptional , Computer Simulation , DNA Copy Number Variations , RNA, Messenger/metabolism , Stochastic Processes
4.
PLoS One ; 11(3): e0149909, 2016.
Article in English | MEDLINE | ID: mdl-26930199

ABSTRACT

Modeling stochastic behavior of chemical reaction networks is an important endeavor in many aspects of chemistry and systems biology. The chemical master equation (CME) and the Gillespie algorithm (GA) are the two most fundamental approaches to such modeling; however, each of them has its own limitations: the GA may require long computing times, while the CME may demand unrealistic memory storage capacity. We propose a method that combines the CME and the GA that allows one to simulate stochastically a part of a reaction network. First, a reaction network is divided into two parts. The first part is simulated via the GA, while the solution of the CME for the second part is fed into the GA in order to update its propensities. The advantage of this method is that it avoids the need to solve the CME or stochastically simulate the entire network, which makes it highly efficient. One of its drawbacks, however, is that most of the information about the second part of the network is lost in the process. Therefore, this method is most useful when only partial information about a reaction network is needed. We tested this method against the GA on two systems of interest in biology--the gene switch and the Griffith model of a genetic oscillator--and have shown it to be highly accurate. Comparing this method to four different stochastic algorithms revealed it to be at least an order of magnitude faster than the fastest among them.


Subject(s)
Computer Simulation , Models, Genetic , Stochastic Processes , Systems Biology , Algorithms , Animals , Gene Expression Regulation , Gene Regulatory Networks , Humans , Models, Biological , Promoter Regions, Genetic , RNA, Messenger/genetics , Systems Biology/methods
5.
J Math Biol ; 72(1-2): 157-69, 2016 Jan.
Article in English | MEDLINE | ID: mdl-25840518

ABSTRACT

We propose a fast and accurate method of obtaining the equilibrium mono-modal joint probability distributions for multimeric systems. The method necessitates only two assumptions: the copy number of all species of molecule may be treated as continuous; and, the probability density functions (pdf) are well-approximated by multivariate skew normal distributions (MSND). Starting from the master equation, we convert the problem into a set of equations for the statistical moments which are then expressed in terms of the parameters intrinsic to the MSND. Using an optimization package on Mathematica, we minimize a Euclidian distance function comprising of a sum of the squared difference between the left and the right hand sides of these equations. Comparison of results obtained via our method with those rendered by the Gillespie algorithm demonstrates our method to be highly accurate as well as efficient.


Subject(s)
Models, Biological , Probability , Systems Biology , Algorithms , Dimerization , Mathematical Concepts , Nonlinear Dynamics , Nucleic Acid Conformation , RNA, Messenger/chemistry , RNA, Messenger/genetics , RNA, Messenger/metabolism , Stochastic Processes
6.
PLoS One ; 10(7): e0134239, 2015.
Article in English | MEDLINE | ID: mdl-26226505

ABSTRACT

One of the functions of the cell nucleus is to help regulate gene expression by controlling molecular traffic across the nuclear envelope. Here we investigate, via stochastic simulation, what effects, if any, does segregation of a system into the nuclear and cytoplasmic compartments have on the stochastic properties of a motif with a negative feedback. One of the effects of the nuclear barrier is to delay the nuclear protein concentration, allowing it to behave in a switch-like manner. We found that this delay, defined as the time for the nuclear protein concentration to reach a certain threshold, has an extremely narrow distribution. To show this, we considered two models. In the first one, the proteins could diffuse freely from cytoplasm to nucleus (simple model); and in the second one, the proteins required assistance from a special class of proteins called importins. For each model, we generated fifty parameter sets, chosen such that the temporal profiles they effectuated were very similar, and whose average threshold time was approximately 150 minutes. The standard deviation of the threshold times computed over one hundred realizations were found to be between 1.8 and 7.16 minutes across both models. To see whether a genetic motif in a prokaryotic cell can achieve this degree of precision, we also simulated five variations on the coherent feed-forward motif (CFFM), three of which contained a negative feedback. We found that the performance of these motifs was nowhere near as impressive as the one found in the eukaryotic cell; the best standard deviation was 6.6 minutes. We argue that the significance of these results, the fact and necessity of spatio-temporal precision in the developmental stages of eukaryotes, and the absence of such a precision in prokaryotes, all suggest that the nucleus has evolved, in part, under the selective pressure to achieve highly predictable phenotypes.


Subject(s)
Cell Nucleus/metabolism , Eukaryotic Cells/metabolism , Karyopherins/metabolism , RNA, Messenger/metabolism
7.
PLoS One ; 9(3): e90285, 2014.
Article in English | MEDLINE | ID: mdl-24594656

ABSTRACT

The development of accurate and reliable dynamical modeling procedures that describe the time evolution of gene expression levels is a prerequisite to understanding and controlling the transcription process. We focused on data from DNA microarray time series for 20 Drosophila genes involved in muscle development during the embryonic stage. Genes with similar expression profiles were clustered on the basis of a translation-invariant and scale-invariant distance measure. The time evolution of these clusters was modeled using coupled differential equations. Three model structures involving a transcription term and a degradation term were tested. The parameters were identified in successive steps: network construction, parameter optimization, and parameter reduction. The solutions were evaluated on the basis of the data reproduction and the number of parameters, as well as on two biology-based requirements: the robustness with respect to parameter variations and the values of the expression levels not being unrealistically large upon extrapolation in time. Various solutions were obtained that satisfied all our evaluation criteria. The regulatory networks inferred from these solutions were compared with experimental data. The best solution has half of the experimental connections, which compares favorably with previous approaches. Biasing the network toward the experimental connections led to the identification of a model that is only slightly less good on the basis of the evaluation criteria. The non-uniqueness of the solutions and the variable agreement with experimental connections were discussed in the context of the different hypotheses underlying this type of approach.


Subject(s)
Drosophila/genetics , Gene Regulatory Networks , Models, Genetic , Multigene Family , Muscles/embryology , Animals , Gene Expression Profiling
8.
J Theor Biol ; 354: 113-23, 2014 Aug 07.
Article in English | MEDLINE | ID: mdl-24632443

ABSTRACT

Cell systems consist of a huge number of various molecules that display specific patterns of interactions, which have a determining influence on the cell׳s functioning. In general, such complexity is seen to increase with the complexity of the organism, with a concomitant increase of the accuracy and specificity of the cellular processes. The question thus arises how the complexification of systems - modeled here by simple interacting birth-death type processes - can lead to a reduction of the noise - described by the variance of the number of molecules. To gain understanding of this issue, we investigated the difference between a single system containing molecules that are produced and degraded, and the same system - with the same average number of molecules - connected to a buffer. We modeled these systems using Ito stochastic differential equations in discrete time, as they allow straightforward analytical developments. In general, when the molecules in the system and the buffer are positively correlated, the variance on the number of molecules in the system is found to decrease compared to the equivalent system without a buffer. Only buffers that are too noisy themselves tend to increase the noise in the main system. We tested this result on two model cases, in which the system and the buffer contain proteins in their active and inactive state, or protein monomers and homodimers. We found that in the second test case, where the interconversion terms are non-linear in the number of molecules, the noise reduction is much more pronounced; it reaches up to 20% reduction of the Fano factor with the parameter values tested in numerical simulations on an unperturbed birth-death model. We extended our analysis to two arbitrary interconnected systems, and found that the sum of the noise levels in the two systems generally decreases upon interconnection if the molecules they contain are positively correlated.


Subject(s)
Models, Biological , Signal-To-Noise Ratio , Stochastic Processes
9.
PLoS One ; 7(11): e47256, 2012.
Article in English | MEDLINE | ID: mdl-23144809

ABSTRACT

Turning genes on and off is a mechanism by which cells and tissues make phenotypic decisions. Gene network motifs capable of supporting two or more steady states and thereby providing cells with a plurality of possible phenotypes are referred to as genetic switches. Modeled on the bases of naturally occurring genetic networks, synthetic biologists have successfully constructed artificial switches, thus opening a door to new possibilities for improvement of the known, but also the design of new synthetic genetic circuits. One of many obstacles to overcome in such efforts is to understand and hence control intrinsic noise which is inherent in all biological systems. For some motifs the noise is negligible; for others, fluctuations in the particle number can be comparable to its average. Due to their slowed dynamics, motifs with positive autoregulation tend to be highly sensitive to fluctuations of their chemical environment and are in general very noisy, especially during transition (switching). In this article we use stochastic simulations (Gillespie algorithm) to model such a system, in particular a simple bistable motif consisting of a single gene with positive autoregulation. Due to cooperativety, the dynamical behavior of this kind of motif is reminiscent of an alarm clock - the gene is (nearly) silent for some time after it is turned on and becomes active very suddenly. We investigate how these sudden transitions are affected by noise and show that under certain conditions accurate timing can be achieved. We also examine how promoter complexity influences the accuracy of this timing mechanism.


Subject(s)
Algorithms , Gene Regulatory Networks , Models, Genetic , Computer Simulation , Humans , Stochastic Processes
10.
BMC Res Notes ; 5: 46, 2012 Jan 19.
Article in English | MEDLINE | ID: mdl-22260205

ABSTRACT

BACKGROUND: This paper lies in the context of modeling the evolution of gene expression away from stationary states, for example in systems subject to external perturbations or during the development of an organism. We base our analysis on experimental data and proceed in a top-down approach, where we start from data on a system's transcriptome, and deduce rules and models from it without a priori knowledge. We focus here on a publicly available DNA microarray time series, representing the transcriptome of Drosophila across evolution from the embryonic to the adult stage. RESULTS: In the first step, genes were clustered on the basis of similarity of their expression profiles, measured by a translation-invariant and scale-invariant distance that proved appropriate for detecting transitions between development stages. Average profiles representing each cluster were computed and their time evolution was analyzed using coupled differential equations. A linear and several non-linear model structures involving a transcription and a degradation term were tested. The parameters were identified in three steps: determination of the strongest connections between genes, optimization of the parameters defining these connections, and elimination of the unnecessary parameters using various reduction schemes. Different solutions were compared on the basis of their abilities to reproduce the data, to keep realistic gene expression levels when extrapolated in time, to show the biologically expected robustness with respect to parameter variations, and to contain as few parameters as possible. CONCLUSIONS: We showed that the linear model did very well in reproducing the data with few parameters, but was not sufficiently robust and yielded unrealistic values upon extrapolation in time. In contrast, the non-linear models all reached the latter two objectives, but some were unable to reproduce the data. A family of non-linear models, constructed from the exponential of linear combinations of expression levels, reached all the objectives. It defined networks with a mean number of connections equal to two, when restricted to the embryonic time series, and equal to five for the full time series. These networks were compared with experimental data about gene-transcription factor and protein-protein interactions. The non-uniqueness of the solutions was discussed in the context of plasticity and cluster versus single-gene networks.


Subject(s)
Drosophila melanogaster/genetics , Gene Expression Profiling , Gene Expression Regulation, Developmental , Nonlinear Dynamics , Algorithms , Animals , Cluster Analysis , Drosophila melanogaster/embryology , Drosophila melanogaster/growth & development , Evolution, Molecular , Gene Regulatory Networks , Models, Genetic , Oligonucleotide Array Sequence Analysis
11.
PLoS One ; 6(12): e27948, 2011.
Article in English | MEDLINE | ID: mdl-22194799

ABSTRACT

Available DNA microarray time series that record gene expression along the developmental stages of multicellular eukaryotes, or in unicellular organisms subject to external perturbations such as stress and diauxie, are analyzed. By pairwise comparison of the gene expression profiles on the basis of a translation-invariant and scale-invariant distance measure corresponding to least-rectangle regression, it is shown that peaks in the average distance values are noticeable and are localized around specific time points. These points systematically coincide with the transition points between developmental phases or just follow the external perturbations. This approach can thus be used to identify automatically, from microarray time series alone, the presence of external perturbations or the succession of developmental stages in arbitrary cell systems. Moreover, our results show that there is a striking similarity between the gene expression responses to these a priori very different phenomena. In contrast, the cell cycle does not involve a perturbation-like phase, but rather continuous gene expression remodeling. Similar analyses were conducted using three other standard distance measures, showing that the one we introduced was superior. Based on these findings, we set up an adapted clustering method that uses this distance measure and classifies the genes on the basis of their expression profiles within each developmental stage or between perturbation phases.


Subject(s)
Gene Expression Regulation, Developmental , Oligonucleotide Array Sequence Analysis/methods , Statistics as Topic , Animals , Cluster Analysis , Gene Expression Profiling , Time Factors
12.
Syst Synth Biol ; 5(1-2): 33-43, 2011 Jun.
Article in English | MEDLINE | ID: mdl-21949674

ABSTRACT

UNLABELLED: Coexpression of genes or, more generally, similarity in the expression profiles poses an unsurmountable obstacle to inferring the gene regulatory network (GRN) based solely on data from DNA microarray time series. Clustering of genes with similar expression profiles allows for a course-grained view of the GRN and a probabilistic determination of the connectivity among the clusters. We present a model for the temporal evolution of a gene cluster network which takes into account interactions of gene products with genes and, through a non-constant degradation rate, with other gene products. The number of model parameters is reduced by using polynomial functions to interpolate temporal data points. In this manner, the task of parameter estimation is reduced to a system of linear algebraic equations, thus making the computation time shorter by orders of magnitude. To eliminate irrelevant networks, we test each GRN for stability with respect to parameter variations, and impose restrictions on its behavior near the steady state. We apply our model and methods to DNA microarray time series' data collected on Escherichia coli during glucose-lactose diauxie and infer the most probable cluster network for different phases of the experiment. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s11693-011-9079-2) contains supplementary material, which is available to authorized users.

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