Stochastic models of chemical kinetics 5. Poisson process

... a single strand of RNA is synthesized using a double stranded DNA molecule as a template. The two strands of the DNA molecule are separated from one another, exposing the nitrogenous bases. Only one strand is actively used as a template in the transcription process, this is known as the sense strand, or template strand. The RNA sequence that is made is a direct copy of the nitrogenous bases in the sense strand. If an Guanine (G) base is part of the sequence on the sense DNA strand, then the RNA molecule has a Cytosine (C) base added to its sequence at that point. In the RNA molecule uracil (U) substitutes for Thymine (T)

... is catalyzed by an enzyme called RNA polymerase, which needs as substrates double stranded DNA, and the ribonucleotides ATP, UTP, CTP and GTP. One at a time, this enzyme adds ribonucleotides to a growing RNA strand by joining incoming ribonucleotide triphosphates to the ribose sugar molecule of the last nucleotide of the growing RNA strand. Two of the phosphate groups are removed from the triphosphate and a covalent bond is formed between the remaining phosphate and the third carbon atom of the ribose sugar at the end of the RNA strand. Initiation of the transcription process begins with binding of the RNA polymerase enzyme to the DNA molecule at a region known as the promoter site. This site is right in front of a gene where transcription will begin. The RNA polymerase enzyme does not copy the promotor into the RNA, but begins the synthesis of the RNA at a specific nucleotide sequence called the start signal or initiation site which is often the bases GTA on the DNA (which then become the bases CAU on the RNA molecule).

Model of transcription: Poisson process time 0

Number of events in time t

Distribution of times between successive events

Simulating the Poisson process (1)

Simulating the Poisson process (2) 0 1 u t

More complex model: birth-death process Transcripts are generated according to the Poisson process, with intensity Transcripts are degraded with the rate constant,   We are interested in simulating the dynamics of this system Start with a number of molecules and advance the state of the system Need to determine: the time of the next event and the type of the event In this system: two types of reactions: generation and degradation

Survival of a state with n molecules

Lifetime of a state with n molecules

Choosing the event (1)

Choosing the event (2)

Simulation algorithm (1)

Simulated trajectories

Probability Distribution at different times

Simulation algorithm (2) 1.Can be generalized for an arbitrary number of reactions and species 2.Use it when master equation is difficult to solve 3.Many extensions for computational efficiency and spatial effects 4.Whenever you have a code, test it using analytically solvable problems Gillespie, D.T. Exact stochastic simulation of coupled chemical reactions. JOURNAL OF PHYSICAL CHEMISTRY, 1977 Vol: 81(25) pp: (read! This citation classic has a very good description of the algorithm)

Gene expression model: Master equation

First moment

Distribution function

Experimental example

Visualizing mRNA molecules in cells (A) Schematic depicting the mRNA detection method. Multiple fluorescent probes bind to each mRNA molecule, yielding a bright, localized signal. (B) Merged image of a three-dimensional stack of images from a CHO cell. Each spot corresponds to a single mRNA molecule. (C) Identification of the spots in the three-dimensional image stack in (C). Each particle found by the image-analysis algorithm is colored differently, showing that the algorithm is accurate and that individual molecules are uniquely identified. The scale bars are 5 μm long.

mRNA copy numbers in cells Histograms showing the distribution of mRNA molecules per cell for three levels of gene activity.