Introduction to Biological Systems
Modeling Biological Networks
Lecture by Professor Jun S. Song
Wiki by Da Lin and Allen Cheng
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Introduction
As of Nov. 2006, a Pub-Med search for the term "genome-wide" returns 6786 hits, all but 38 of which postdate 1995, the year in which microarray technology was first applied to gene expression research.
Before the 1990s, the popular method of conducting mutation and gene expression analysis operated on a "one-gene-one-experiment" basis, which studied individual genes and proteins using northern blot, southern blot, and gel shift experiments.
This experimental framework guided researchers towards three main questions which focused on the structure of gene transcription:
1. Which genes are highly expressed?
2. What are the factors regulating a gene?
3. Where does transcription start?
The publication of "Quantitative monitoring of gene expression patterns with a complementary DNA microarray" in 1995 by Schena et al. became the first step in revolutionizing the study of gene expression. With the introduction of microarrays, researchers can now access comprehensive measurements of the expression levels of hundreds of genes simultaneously, allowing for the analysis of the network of protein/gene interactions.
With this shift in technology came a shift in the focus of gene expression research, which now focuses on the connectiveness and relationships between each component:
1. How are genes regulated?
2. How are genes interacting?
3. Why are genes interacting in a certain way? Is it for robustness or for stability?
4. How and why does a genetic network break down - or rather, why and how are diseases caused?
These questions drive the emerging field of Systems Biology, which views organisms as as an integrated and interacting network of genes, proteins and biochemical reactions. On this page, we will be introduced to basic dynamical systems and local stability analysis. Dynamical systems have been extensively developed in both physics and chemistry, but is still an emerging method in biology. Nonetheless, as we shall see, dynamical systems have already shifted the way biological systems are perceived and studied.
Systems Biology and Dynamical Interactions
Traditional biology often perceives biological organisms as a complex system which can be analyzed in terms of the properties of its parts. However, such methods can only yield limited insights into biological systems. Systems biology approaches living organisms as integrated dynamics wholes whose properties cannot be reduced to smaller parts. For example, the immune system cannot be studied by analyzing the properties of individual genes or mechanisms. Rather, the immune response be understand by observing the interactions of numerous genes, proteins, and niches which work together to produce the immune response.
Taking another example, if one wished to study the automobile, and closely examined the engine, the muffler, and even the speedometer. In the end, one would have close to no idea how the automobile runs, and more importantly, one would not know how to fix the car when it breaks down. 
In the same fashion, traditional biology has provided very little understanding of how organisms operate and how to prevent and cure complex diseases, such as HIV and cancer.
Biological studies over the last century have revealed that the foundational components of biological activity – genes and proteins – almost never operate alone. Like that of an automobile, the interactions are often complex yet structured, sometimes conserved across multiple species. Furthermore, researchers have realized that analysis of the biological system at various levels of organization reveals different insights on the behavior of each of the foundational components.
Systems biology is thus the study of dynamical interactions which govern the genetic network, the evolution of RNA or proteins. The laws of interactions acting upon this system are given by chemical kinetic equations derived from empirical observation of DNA-protein and Protein-protein interactions. Quantitative modeling of the genetic network is founded on enzyme kinetics, assuming the transcription factors as substrates and DNA as enzymes. Robust regulation and stability factors acts as components of the feedback control loops, and phenotypes become the equilibrium states - or the stable steady states. In other words, systems biology studies biological systems as a dynamical system of interactions.
Examples of Dynamical Systems
Models of dynamic systems are often used to model economic, physical, chemical, and environmental trends. Examples of dynamical systems include the swinging of a clock pendulum, the flow of water in a pipe, the number of fish each spring in a lake, or the mathematical lorenz attractor. Below we will be examining two of the most commonly used dynamical systems in science: simple harmonic motion of a spring and chemical rate of reaction.
While we go through the examples, keep a lookout for these common themes in dynamical systems:
1. Differential equations reduce to algebraic equations at equilibrium.
2. We are often interested in non-equilibrium but quasi-steady state approximations (locally stable points)
3. We need to know which equations can be approximated to zero, based on the time scales of kinetics (slowly varying rates can be set to zero).
Harmonic Oscillators
Chemical Kinetics
Michaelis-Menten Eqn.
We now apply the mass action law to the physical process of transcriptional regulation.
Transcription is the process by which RNA Polymerase attaches to DNA and creates a complementary strand of ribonucleic acid using DNA as a template. The RNA Polymerase often requires association with transcription factors, which aid the Polymerase enzyme through the entire process from initiation to termination.
Where S and E represented Substrate and Enzyme in the mass action law, we now represent S as the transcription factor, DNA as the enzyme. We also introduce a secondary equation in which the substrate-enzyme complex produce a dissociated product, the RNA strand, and enzyme, or DNA.
The given association of the terms with their corresponding biological molecules may seem counter-intuitive: DNA is not the actual enzyme catalyzing the reaction, nor is the transcription factor the substrate. In truth, the RNA Polymerase and transcription factors impart enzymatic activity. Moreover, it may appear that by allowing S to represent the transcription factor, we allow it to disappear from the equation or form the product. In actuality, the transcription factor is intact after the transcription reaction.
We can address these issues with two counterpoints: first, this is a gross simplification of the true transcription reaction, which involves many more enzymes and substrates. A series of equations representing the true half-lives of all protein components in the pathway is outside the scope of this example. Secondly, the in vivo equivalent of disappearance of transcription factor would be protein denaturation, which occurs with time. The attribution of DNA to the enzyme component of the equation signifies the relative stability of DNA compared to proteins. Finally, keep in mind that the general principle of the equation matches empirically.
From the reaction equation presented, we can write four equilibrium equations:
We hope to eliminate the equations to as few variables as possible, so we employ the mass action law:
and eliminate the E from the top three equations, yielding two equations to solve:
We can solve the two differential equations by assuming quasi-steady state conditions. For this, we will chart two time scales:
Time 1 is thus substantially less than Time 2 - that is, SE is formed much more quickly than product appears. Biologically, this would mean that the association of RNA Polymerase to DNA occurs more quickly than the transcription of hundreds of nucleotides, a reasonable argument.
The above equation models the saturation effect – because S>>E, SE is limited.
A summary of the equations and assumptions for the basic Michaelis-Menten Equation is now presented:
Equation Modifications
Hill Cooperativity
Transcriptional regulation requires the activity of many proteins working in concert. In the image we see a number of transcription factors associating with RNA Polymerase before transcription even proceeds.
We now increase complexity by adding n substrates into the Michaelis-Menten equation.
The equation assumes that the substrates all bind simultaneously, though it actually approximates fast sequential binding. n is the Hill coefficient and describes the degree of cooperativity in enzyme-multi-substrate formation. Once again S = transcription factor, E = DNA, and P = RNA.
Catalytic Activators/Independent Cofactors
We introduce an activator A, which binds independently of the substrate. The in vivo counterpart would involve the binding of two independent transcription factors that bind to the promoter of the gene, both of which are required to initiate transcription.
To solve these equations, we assume that concentrations of SE, SEA, and Etotal are constant for steady state reasons. The corresponding derivatives are then 0.
Specific Activation
We now involve the same activator and transcription factor, but we now assume that they bind sequentially – that is, the activator recruits the transcription factor for binding.
We then yield the equation:
Competitive Inhibition
We introduce an inhibitor I, which prevents the enzyme from participating in the reaction. The in vivo equivalent would involve a repressor and transcription factor that bind at the same site.
Yielding:
Noncompetitive Inhibition
Substrates and inhibitors can also bind at different sites so that both can bind to the enzyme simultaneously. However, only SE is active, meaning that the presence of an inhibitor regardless of substrate prevents the enzyme from participating in the reaction.
Uncompetitive Inhibition
Inhibitors can also bind to enzymes only after substrates. This would form SEI. Only when the inhibitor is removed can the reaction proceed.
Note to STAT115 Students: Derivations of these results follow similar reasoning as that used to solve the equations for the first Michaelis-Menten Equation. Details, however, are most likely unnecessary for the course.
Steady State Analysis
Steady States
for a given value of parameter k are the points where f(x,k) =0. Stable steady states correspond to asymptotic states that do not change with time and are robust against small perturbations. A steady-state equilibrium is globally stable if the system converges to the steady-state equilibrium regardless of the level of the initial condition, whereas a steady-state equilibrium is locally stable if there exists a set of the steady-state equilibrium such that for every initial condition within this set the system converges to this steady-state equilibrium.
Example 1
From the graphs, how many steady states are there for each value of k? Which ones are globally stable?
Solution
This differential equation can be solved simply by Mathematica, or a similar math program. Each steady state corresponds the values for x for which f(x,k) has a real zero. If we turn our attention to the graphs, we notice that depending on the value of the parameter k, we obtain different steady states. When k<0, there are no steady states; we k =0, there is one locally stable steady state (blue); when k>0, there are two steady states, one locally stable(blue), the other globally stable(red). When the qualitative behaviors of the solutions change with the parameter k, the system is said to bifurcate. An example of bifurcation in real life can be seen in cell differentiation, such as asymmetric neuroblast division which results in both a ganglion mother cell and another neuroblast.
Example 2
From the graphs, how many steady states are there for each value of k? Which ones are globally stable?
Solution
Like question one, we can solve this differential equation with Mathematica or a similar math program. Turning to the graphs, we can see that there is one globally stable steady state (red) at k>1/4, because a slight disturbance along the x-axis in either direction will return to the equilibrium point. When k = 1/4, there are two steady states, one locally (blue) and the other globally stable (red). When 0The system bifurcates at k = 0, and 1/4. If we look at the plot of x against k, for every value of x, we can see the number and qualitative nature of its corresponding steady states. For example, at k < 0, there are 3 line segments in that region, 2 red and 1 blue, which corresponds to the 2 globally stable and 1 locally stable steady states that we observe at k <0. The points of bifurcation are the points in which there exists an abrupt change in the nature of the equilibrium points.
References and Additional Resources:
Readings
Bonetta L. "Systems biology--the new R&D buzzword?" Nat Med. 2002; 8(4):315-6
Harvard Link
Ideker T, Galitski T, Hood L. "A new approach to decoding life: systems biology." Annu Rev Genomics Hum Genet. 2001;2:343-72.
Harvard Link
Savageau, M.A.: Biochemical Systems Analysis. A Study of Function and Design in Molecular Biology. Addison-Wesley, Reading, Massachusetts, 1976.
Torres, N.V., and E.O. Voit: Pathway Analysis and Optimization in Metabolic Engineering. Cambridge University Press, Cambridge, U.K., 2002.
Voit, E.O.: Canonical Nonlinear Modeling. S-System Approach to Understanding Complexity, Van Nostrand Reinhold, NY, 1991.
Voit, E.O.: Computational Analysis of Biochemical Systems. A Practical Guide for Biochemists and Molecular Biologists, Cambridge University Press, Cambridge, U.K., 2000.
Links
Institute of Systems Biology Link
Systems Biology at Pacific Northwest National Laboratory Link
Wikipedia on Systems Biology Link
References
Stat115 Lecture (Jun. S. Song) Harvard Link
Additional Images from:
"Trends in Complex Systems Biology." Powerpoint by Eberhard O.Voit
Link
Institute of Systems Biology
Link
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