This course will introduce the student to contemporary Systems Biology focused on mammalian cells, their constituents and their functions. Biology is moving from molecular to modular. As our knowledge of our genome and gene expression deepens and we develop lists of molecules (proteins, lipids, ions) involved in cellular processes, we need to understand how these molecules interact with each other to form modules that act as discrete functional systems. These systems underlie core subcellular processes such as signal transduction, transcription, motility and electrical excitability. In turn these processes come together to exhibit cellular behaviors such as secretion, proliferation and action potentials. What are the properties of such subcellular and cellular systems? What are the mechanisms by which emergent behaviors of systems arise? What types of experiments inform systems-level thinking? Why do we need computation and simulations to understand these systems? The course will develop multiple lines of reasoning to answer the questions listed above. Two major reasoning threads are: the design, execution and interpretation of multivariable experiments that produce large data sets; quantitative reasoning, models and simulations. Examples will be discussed to demonstrate “how” cell- level functions arise and “why” mechanistic knowledge allows us to predict cellular behaviors leading to disease states and drug responses.
Mathematical Representations of Cell Biological Systems I
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Do curso por Icahn School of Medicine at Mount Sinai
Introduction to Systems Biology
233 ratings
Icahn School of Medicine at Mount Sinai
233 ratings
Curso 1 de 6 em Specialization Systems Biology and Biotechnology
Na lição
Mathematical Representations of Cell Biological Systems | Simulations of Cell Biological Systems
Module description goes here.
Conheça os instrutores
Ravi Iyengar, PhDDorothy H. and Lewis Rosenstiel Professor
Department of Pharmacology and Systems Therapeutics, Systems Biology Center New York (SBCNY)
[SOUND]
This is lecture five in the Introduction to Systems Biology course.
in, in lectures two, three, and four, I tried to introduce biological systems
in sort of a very simple, some might even call it a Mickey Mouse type level.
To
to physical scientists, mathematicians, and others to see,
what is the systems that require the kind of
mathematical representations to kind of compute build my,
build computation models, run numerical simulations that compute systems behavior.
In lectures 5 and 6, I'm going to turn around and try to
describe to biologists who might not be
so familiar with computational methodologies and approaches.
What is it that computation can help us do to understand systems
in a deeper fashion?
Like the previous ones which sort of skimmed over
many, many details, each lecture could have been sort
of a full course, the topic that I covered
in each lecture could have been a full course.
This is also true in lectures five and six.
These two lectures can actually form the basis for a full course,
and indeed do in many, many applied math programs.
So how do mathematical representations help us solve biological problems.
What mathematical representations do is to deal
with complex systems in an orderly fashion.
And in the case of cell biological and regulatory biology problems, allow us
to predict IO or, or, or input output relationships
as a function of time or space, or other variables.
So I've taken this cartoon from a very old review that I wrote with a couple
[UNKNOWN] in my lab back in 1999, in
a series that Science did on complex systems.
And it talks about what is the basis
of complexity in cell biology or cell signaling system.
The basis of complexity is just
the multiplicity of components and interaction.
So even in a very simple pathway such as the sapture
going to G protein, going to an effector like Adenylate cyclase.
One would say that if you want understand the
relationships between receptor occupancy and the activation of Adenylate cyclase.
While you could dial up a nice model to
understand this and this will give you all the
relative relationships between R and G and G and E
to to give you sort of a full in depth understanding.
But you could say, well I could also do
experiments by varying the amount of lag in the
atom, looking at varying the amount of measuring the
change in activity of Adenylate cyclase or production of cytokine.
And indeed this has been done a lot, for a long time, and you can sort
of get an empirical understanding, without casting the results in sort of
a [UNKNOWN] computational model.
Although this may work at the level of individual pathways, it certainly
breaks down at the level when you have two interacting pathways as shown in B.
Or multiple interacting pathways with some trans-location of the eh, trans-location
of the eh, signaling components from one compartment to another.
As in this case where signals from the membrane might go into the
nucleus or signals from the cytoplasm to what can go into the nucleus.
So that these kinds of systems even with small number of components we can see that
a, even just two pathways one has 11 interactions and 22 rate constants.
With these sort of multi-component system, one has a total of, in the cartoon I've
showed here, at 70, nearly 70 interactions, and 138 rate constants.
All of these, allow us to sort of say that well,
if you don't really have a model, how can we understand in
such a system, when a signal comes at a receptor at
this level and just causes a transcription factor to be a, activated.
In the nucleus, in the production of a
protein p in response to the transcription factor activation.
This relationship in quantitative terms would be nearly
impossible to understand and actually is impossible to understand.
You can always sort of plot out empirical relationships.
Again, you can add enough ligand, varying
amounts of ligand to a cellophane [INAUDIBLE]
and measure the production of this protein,
or the activation of this transcription factor.
And plot this as a function of time,
and the experimental literature is replete with such examples.
But that doesn't tell you what the, nature of these relationships are, or
allow you to predict how the experiment might work in another circumstance.
Unless you repeat the experiment
very similarly as was done previously.
So if one believes that one requires mathematical representation
or of one, except that one requires mathematical representation.
And computation to understand such complex systems, we can
ask the question, how are these mathematical representations allow us
to understand, or what sort of operations do we need to do on these
mathematical representation to get us an understanding of the systems behavior?
There are two types of there are two types of
if you want to call it solutions.
One is called, the first one is called analytical solutions.
And an analytical solution is, is an equation of
a set of equations, that has a solution that
can be expressed as a mathematical expression in terms
of well-known, symbolic operations that can be used for calculations.
So, for example, just a very simple equation
that many of you have seen in high
school algebra and it has an analytical x
squared plus bx plus C, has an analytical solution.
Shown here which allows you to compute the value of x.
So in respective of values of a, b, and c, this relationship
shown on the right hand side will always define the value of x.
There are very few examples of such analytical
solutions and actually a, a, in the previous lecture
using the austrian meggian model and the hodgkin huxley
models I've shown you some of the best known.
Or widely accepted analytical models in Biology
they are very very few so just don't
take because I used this two examples, don't
think that this is commonly achievable in biology.
Most of the time the complexity
of the biological systems preclude analytical solutions.
And so what people do is to go to numerical solutions.
Numerical solutions or numerical analysis involve giving numerical values
to solve the equations and solving these numerical equations equations
numerically allows you to understand input/output relationships,
processing of information within the system, and so on and so forth.
So how are numerical solutions obtained, or how are models cast for numerical
solutions is what we will consider in this set of a slide.
So these slides that I am describing are mostly with
words and without any, any interested cartoons or pictures to look at.
But please don't think that because it's wordy and
you don't pay attention to what, what I'm saying here.
The points I'm making here are very important in terms of understanding
how to apply models and build useful and good models in biology.
There is a large literature where physicists and
mathematicians built models for a long, long time, which biologists
basically rolled their eyes at, and thought, what is
the use of this, and were not useful in biology.
So I think it's very important in terms of thinking about computational models to
solve biological problems is that one build
good biological models that actually solve biological problems.
So one of the important criteria here is that
these models need to be relevant to the experimentalist.
With the coming generation of students fireside and both experimentation.
And in mathematical and quantitative reasoning, the
experimental list and the modeller, and the
modeller will be the same person, so they can always think in an integrated manner.
Good model need to generate to provide answers
and generate hypotheses that cannot be intuitively obtained.
Intuition is just another word for our own mental computation.
If we can look at a certain observation and sort of mentally compute or
calculate what that observation means and how
it might predict how the system might behavior.
There is no particular advantage in sort of say writing out equations,
putting them to numbers to them and calculating to see how the systems behave.
You can just think about it intuitively in your mind
and predict the behavior and use the prediction from the behavior.
They use the predictions that you think about to dial up new experiments to test.
So, good models that require computation should produce,
generate hypothesis that cannot be intuitively obtained.
And I'll give you some examples of that.
And it's also really important that models do not oversimplify.
Cell biology's all about details and these details
are very important in understanding how normal systems work.
How they're perturbed in disease or pathological conditions.
And when one oversimplifies or when model parameters cannot be
explicitly ascribed to experimentally observable functions,
such models typical, are generally ignored by experimental biologists.
And generally have very low value in terms
of understanding at a mechanistic or deep level
how the organization of the system leads to
the sort of a behavior that is observed.
So oversimplification of molecular details obscured biological
mechanisms, that is how a biological phenomenon occurs.
So again but if you put in everything in the kitchen sink, it just serves to
confuse the, the system, and, sort of also
can obscure developing a deep understanding of the system.
So, getting the right balance of detail is
a very critical aspect in developing a good model.
And this comes from actually trial and error from the system of interest.
And as we build more and more models over the
period of the next few decades I suspect that rules
will emerge by which were the different, what is the
right model of detail required for the question of interest.
So the key point here is that good models make non-intuitive
predictions that actually require computation to generate the prediction.
The most common type of models, used in, cell biological and regulatory
biology are models that are based on ode's or ordinary differential equation models.
Here is an example of, here is a cartoon from a system from a
review I wrote with one my former post-doc, John Hume Demeron back in 2004.
That shows how one can write this path,
this two pathways that go from norepinephrine to sort of persistent
MAP kinase activity in terms of the set of ordinary differential equations.
Here it looks, you can look at the control all the way from the all the way
from norepinephrine binding at the receptors to the activation of MAP kinase.
This cartoon here allows us to sort of make two points that are quite important.
One point is that the level of detail that is required.
One could say that instead of having two kinds of receptors, one could have just
used the term adrenergic receptor that sort of subsumes norepinephrine
binding to the alpha-1 and the alpha-2 adrenergic receptor.
If one had done that, you can see that
it might erroneously make us think that the same
receptor could activate both the Gq and a GI
GO pathways leading to this feedback loop shown out here.
But in reality there are two distinct gene products,
the alpha-1 adrenergic receptor and the alpha-2 adrenergic receptor.
Both are which bind the same ligand actually with
rather similar affinities to activate entirely different signaling pathways.
So this kind of detail that says that I
need to have receptor isoforms, the alpha-1 and alpha-2 as
distinct proteins or distinct components or reactions in my model
is a very important aspect of making the model realistic.
Once one does that, one can then see that how
connectivity in the model can lead to a feedback loop.
And you can ask the question, what is the, a value of this feedback loop out here?
In in the, in how not epinephrine at the cell surface receptor
causes persistent MAP kinase activity that can lead to various changes of cell.
So you can say that the research question and in the models such as
this says, why does norepinephrine, norepinephrine cause
persistent MAP kinase activities in some neurons?
The, as I mentioned before, the detail required here is that the
different isoforms, the receptors and their
pathways, need to be explicitly represented.
And the numerical analysis can tell us when
this positive feedback loop out here, can function as
a switch such that, at a certain concentration
of norepinephrine, you can get persistent map kinase activity.
The advantages for ordinary differential equation
models are that ODEs are typically
reasonable representation of a very large
number of biochemical reactions within the cell.
Kinetic parameters for ODEs can be often
explicitly measured or estimated from time course experiments.
There computer programs such as [UNKNOWN] and other [UNKNOWN] now
that allow for this kind of estimation so this is sort of a do able thing.
There are often substantial experimental data such as times course data, times
course data that can be used to build well constrained ODE models and people.
Suddenly people by up, without experience in
biology, tend to think that oh, there are
many components, its like a model that
cannot be constrained and can show any behavior.
This is actually not true, the time course data can be used
to build well constrained models and we have done this many many times.
Having for different kinds of systems.
And there are of course many software suites such as Matlab
and Octave and others that can be used to solve these ODE models.
So solving ODE models is now very I won't say
easy but feasible approach for many people to use.
ODE models have limitations one of them,
and many of these are conceptual limitations.
ODE models are built using the well stirred reaction, reactor assumption.
Which means that all reactants are presumed to have
equal access to all other reactants without any hindrance.
Many times this is true.
But sometimes there is due to the presence of scaffolds and anchoring
proteins and others, this kind of, well stirred assumptions are not correct.
So this can lead to error in the modeling.
So the well stirred assumptions, sort of work at the level of the cytoplasm.
But not when, say, cytoplasm and membrane, or cytoplasm and nucleus are involved.
So this is something to constantly think about
in developing the model, in such a model.
And most often, interactions between
multiple, entities, multiple entities such as
protein kinases and transcription factors,
involve different compartments or multiple compartments.
And so the, well stirred assumption of a single compartment
where everything can interact with everything else is not valid.
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