Table of Index

  1. 1.

    GAL System: A brief overview

  2. 2.

    Genetic Components of GAL system

  3. 3.

    Transcriptiona and regulation

  4. 4.

    Special characteristics

  5. 5.

    Long Term Adaptation

  6. 6.

    Ultrasensitivity

  7. 7.

    Hysteresis

  8. 8.

    Hysteresis, affecting factors

  9. 9.

    Feedback loops

  10. 10.

    Noise and stochasticity

  11. 11.

    The Gillespie algorithm

  12. 12.

    Implications of Stochasticity in GAL system

  13. 13.

    LTA

  14. 14.

    Hysteresis

  15. 15.

    Objective

1.GAL System: A brief overview

Galactose metabolism is crucial in almost all known life forms from prokaryotes to mammals. Saccharomyces cerevassie has been chosen as the model organism for various obvious reasons, including the ease of culturing, low replication time and similarity of yeast genetic system to prokaryotes and higher eukaryotes. The galactose metabolism system or GAL system of yeast is a system of enzymes that are responsible for the uptake of galactose into the cell and convert the galactose into glucose-6-p which can in turn enter the glycolysis pathway and further be metabolised into energy and other metabolites such as amino acids. The metabolic pathway is known as the Lelior pathway, named after the famous biochemist Luis Federico Leloir who, with his team, elucidated the basic mechanism of galactose metaolism in 1948 and later won the Nobel prize for chemistry(1970) for his contribution to biochemistry in elucidating lactose and related metabolic pathways(Holden 2003). The pathway involves the action of 4 enzymes, namely galactose mutarotase facilitates the conversion of β-D-galactose to α-D-galactose, galactokinase that converts galactose to galactose-1-phosphate, D-galactose-1-phosphate uridylyltransferase converts galactose 1-phosphate to UDP-galactose using UDP-glucose as the uridine diphosphate source and UDP-galactose 4-epimerase recycles the UDP-galactose to UDP-glucose for the transferase reaction. The detailed schematic of the reaction is give below /home/kishore/Desktop/800px-Leloir_pathway.png Apart from the enzymes involved in the leloir pathway, another enzyme called the galactose permease allows cellular uptake of galactose and hence is a crucial part of the GAL system. The regulatory part of the GAL system consists of 3 proteins, gal4p, gal80p and the gal3p, the details and importance of which will be discussed in detail in the coming sections and these proteins are also the main focus of this study. together, these 9 enzymes comprise is what is referred to as the GAL system.

1.1 Genetic structure of the GAL system

The structural genes of the GAL system, GAL1, GAL7 and GAL10 are concentrated on the chromosome II, with GAL10 and GAL7 on the same strand, and GAL1 on the opposite strand, sharing the promoter with GAL10. The Upstream activation sequence(UAS) of GAL1 and GAL10 is hence common and has 4 binding sites for gal4p(Giniger 1985). The GAL1 codes for galactokinase and GAL10 codes for UDP-galactose 4-epimerase. GAL7 which codes for the D-galactose-1-phosphate uridylyltransferase. Promoter information needs to be added. GAL2 gene which codes for The regulatory genes, GAL4, GAL80 and GAL3 are located on chromosome XVI, XIII and IV respectively. GAL4 is under constitutive expression while GAL3 and GAL80 have UAS with gal4p binding sites.

1.2 Transcription and regulation

The transcriptional regulation of the GAL system is quite complicated. It involves presence of activators, inhibitors, inhibitor of the inhibitor, positive and negative feedback loops and some other elements. The primary element of the transcriptional system is gal4p. As we have seen above, both structural and regulatory genes have gal4p binding sites. gal4p is a 881 residue long protein, making it very difficult to isolate, characterize and study and consequently, it was a long time before the whole protein was characterized. gal4p is the activator of the gal transcription. The expression of gal4p is constitutive and is only regulated via protein degradation. gal4p, when bound to the UAS of the GAL genes, recruits SAGA complex(Chasman 1990)(Larschan 2005) which recruits the TATA binding complex which in turn initiates transcription. Studies indicate that gal4p protects the guanidium nucleotide in the UAS from being methylated and hence inhibition of gal4p binding to the UAS inhibits the expression of these proteins(Giniger 1985). gal4p is sequestered by gal80p.

Noise and Stochasticity

Noise

Noise is commonly is a common phenomenon in experiments. The signal that does not have any role on the observed behaviour of the phenomena in study. Even though noise is generally eliminated from the observations, it is not completely useless. Presence of noise signals that the system is on, and the study of the same noise can tell one whether a particular set of results are biased due to experimental or instrumental conditions. As is evident from every study of any form science, noise is omnipresent. No system can exist without noise and lack of noise thereof indicates that the system is dead.

Statistically, noise is defined as the recognized amount of unexplained variations in a sample. Incidentally, most of the observable noise also has a very low amplitude, which is arises because all systems are composed of numerous smaller systems which act independently, making the noise weak and negligible. In a thermodynamics point of view, noise is crucial because it imparts entropy to the system which in turn makes reaching of that particular state more spontaneous.

The BIG-BANG theory of noise: Going back to the origins of the universe, Big-Bang model is the widely accepted theory on how the universe came to be. The model states that the universe expanded from a high temperature, high density primordial state, which caused rapid decrease in density and temperature, creating fundamental particles in the process. (source: https://www.britannica.com/topic/big-bang-model) Now, let us consider this in a statistical thermodynamics point of view. Formation of a high temperature, high density state implies a huge decrease in the entropy, i.e., an extremely large and negative TΔS term and a positive ΔH term, unless the state comes from a still higher energy state, which is even more unlikely. Hence, the ΔG of such a process is highly positive, making the probability of formation of said state extremely small. Furthermore, the cosmological principle, the implication of which is that the universe has no edges, states that big-bang happened, not at a single point, but throughout the space at the same time, making the probability of its occurrence even lesser. This primordial state, hence, fits perfectly with the definition of noise given earlier. And hence comes the conclusion that the universe was but the product of noise. In a different perspective, if big-bang was not a noise signal, more such big-bangs would be happening which would cause definite instability in the existing universe(s).

Stochasticity Stochasticity, simply put, is the randomness of a system. From the time Robert Boyle documented the idea of chemical reactions in 1660s in his book: The Sceptical Chymist, scientists have been studying the innumerable chemical reactions happening around us in the nature. In the past century, the concepts of chemical reactions have advanced a lot. The concept of chemical equilibrium, chemical kinetics, chemical energetics and arrhenius equation stand at the center of all the chemical phenomenon. These concepts are able to predict the temporal and quantitative details of any chemical reactions when certain parameters are known. However, the prediction is never completely accurate, there are always unexplained variations, in other words, noise. For example, for an reaction where reactant A is converted to reaction B, equilibrium constant should be calculated by taking average of the ratio of steady state concentrations of A and B by conducting the experiment a number of times, in other words, by creating a sample space of equilibrium constants, which is big enough to generalize the obtained mean value. Same goes for kinetic studies. Fortunately, however, for all practical purposes where the reactant and product numbers are always huge (of the order of e18-e23), the result remains the same to the significant digit, which is generally 2nd or the 3rd decimal. However, when the concentrations are very low, for example a cell, and the significant digit moves from 3rd to say 7th digit, variations observed above become too big to predict the system’s behaviour within reasonable error(Quack 1983). The explanation to this lies in the concept of stochasticity. If we consider that the reactions are stochastic, for the reversible reaction A to B, it would mean that at a given point of time, A may convert to B or B to A, with a bias proportional to the rate of each reaction or event. Hence, if we sample enough reactions, it is possible to isolate an instance where all A is converted to B, or all B is converted to A. When a probability distribution of the concentration ratios is plotted, the mean of the distribution will be equal to the so called equilibrium constant.

Role of noise/stochasticity in evolution: The classical set of experiments conducted by M. Delbruck(Luria 1943), where he studied the presence of viral resistance in a culture of bacteria of the same quantity of the cells, originating from a single cell isolated from a colony that was sensitive to viral infection. The theory was, that if the mutation process is stochastic and not affected by the dawn of viral infection, there will definitely be a finite number of viral resistant cells in each culture as viral resistance will have to occur from random mutations, and that this occurrence of the resistant cells will follow a poisson distribution, as expected of a stochastic process. The results had indicated that mutation is a stochastic process. A feature of interest here is that the fraction of the resistant bacterial population is always very less, i.e., a noise in the demographics of the entire population. And that noise itself is responsible for development of antiviral characteristics and hence for survival. Darwin’s theory of evolution, the natural selection theory, states that evolution happens via the selection of small, inherited variations that provide an advantage to the organism over the other variations. Delbruck’s experiments stated above are a good example of the same. And by extrapolation, one can visualize the importance of noise in evolution.

GAL models in the literature

GAL system has been a cryptic puzzle for more than 3 decades and the recent publications concerning the system discovering and exploring the non-coding RNA of GAL10 and it’s impact on various aspects of the switch such as lack of basal expression of galactokinase in absence of galactose, ultrasensitivity and systemic hysteresis(Lenstra 2015) or another study showing the importance of gal4-gal80 interaction strength for the establishment of bistability of the system(Das 2014) as an irrefutable proof for the fact. Many groups have developed models of the GAL system based on the available literature and their own findings to explain certain features of the GAL system such as ultrasensitivity, Long term adaptation, hysteresis etc. A few of them are mentioned here for a general perspective.

Venturelli et.al., 2012 The mathematical model constructed by Venturelli et.al., —(Venturelli 2012)— was based on the assertion that the bimodal behaviour of the GAL switch is caused by dual feedback loops. They studied the behavior of single cells in terms of GAL protein expression profiles by using fluorescent proteins tagged to the GAL10 promoter. By studying the feedback loops via deletion and modulation, they figured out that the gal1p and gal3p feedback loops together cause systemic bimodality and based on this they constructed a deterministic model. The model, however, fails to explain the loss of bimodality when the feedback loops remain intact and the affinity of gal4 to gal80 is changed—(Das 2014)—(need to check once). Another model, which takes into account the cell growth and division processes into account —(Song 2015)— replaces the transcriptional dyanmics of GAL genes with a free parameter that is determined by scanning a range of values for the parameter and fitting the data with experimentally obtained curves. This method fails to provide the intricate details of the transcription dynamics and for every other transcriptional system, this free parameter needs to be determined. Hence, for a new system, it is not clear how much experimental data must be generated to capture the parameter that completely protrays the behaviour of the system. A monte-carlo based stochastic model that considers all the possible interactions of the system has the potential to provide whichever information is needed of the system. And unlike a deterministic model a stochastic model consists of the most basic of the elements and hence it is easy to incorporate new elements into the model that were not known before. It is also possible to determine the effect of every individual parameter on every aspect of the system. The biggest advantage of a stochastic model, however, lies in the fact that it can provide details of single-molecule dynamics over time which allows one to determine the effect of noise on the system, as is in the case of LTA of GAL network. It is also simpler to generalize a stochastic model than it is to generalize a deterministic model.

Objective

The main objective of this project is to culminate all the observations made till now on GAL system into a stochastic model (Greger 2000)— —(Holden 2003)— —(Majumdar 2004)— —(Kar 2014)