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.
Simulations of Cell Biological Systems I
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Из курса от партнера Icahn School of Medicine at Mount Sinai
Introduction to Systems Biology
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Icahn School of Medicine at Mount Sinai
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Курс 1 из 6 — Specialization Systems Biology and Biotechnology
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Mathematical Representations of Cell Biological Systems | Simulations of Cell Biological Systems
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Ravi Iyengar, PhDDorothy H. and Lewis Rosenstiel Professor
Department of Pharmacology and Systems Therapeutics, Systems Biology Center New York (SBCNY)
Welcome to lecture six.
I'm continuing on describing what one can do with
various kinds of models, simulations of different types of models.
Depending on the kind of parameters one uses, one can have Toy Models,
or one can have plausible models a term that
I have made up, or one can have identifiable models.
Toy Models often use arbitrary parameters that may or may not be and most often are
not, later have any relationship with the, parameters in the real systems.
And can be used to make sort
of theoretical points, with regard to the model.
Plausible models typically use experimentally
measured, or es, or estimated values.
The kinetic parameter, such as rate constants, or the levels of
reactants, are, are generally not, cell type specific.
They're either canonical and hence may not be fully representative of every cell
type or the cell type of interest for which the models are being created.
So this, this is kind of a general model, if you want to call it that.
It's likely to be real, but it may not be exactly real, for the cell
type, and so, this is why I'm calling these plausible models.
These are the most common type of dynamical models and systems biology, and
often times the use of canonical
parameters does not really affect the systems,
and this can be experimentally tested, when, when looks at model predictions and
tries to validate these, or sort of test these model predictions in by experiments.
Another class of models which is most common in pharmacology,
are these types of ODE models that are built to explain experimental data.
These are called identifiable models.
These are system-specific, and the models are directly fitted to experimental datas.
These are most commonly used in drug action, and so while the model
parameters are fitted to experimental data and the fitting is quite
it has a level of accuracy that can be statistically characterized.
The model parameters may often not be connected
to molecular details, molecular details so one might not
get a mechanistic understanding of behavior, but one gets
a precise quantitative description of behavior in identifiable models.
So when one goes from mathematical representations to numerical
simulations, what one needs to do is to get the reactions parameterized.
So the initial concentrations and the reaction
rates have to be reaction rates are
unneeded and these are not always easy
to obtained as biochemical and cell biological
experiments a vast majority of these which were done for the last 50 or so years,
had not been really geared towards getting rate measurements or absolute values of
components within cells.
As we get more and more into these kinds
of modeling systems, one, people are starting to collect
these type of data, but still there is very
little of this data that is really available very easily.
They often need, one needs to read experimental papers
and make some assumptions to extract these kind of data.
Sometimes these parameters need to be guesstimated
based on known value for similar parameters, in
one where you can think of there's a
little bit like homology modeling of structures, where
if you know, one protein structure then a protein, an ISO forming a, a have
similar structures sort of based, and you think
sort of calculate the structure based on computation
similarly if one has values for say, GQ, one could, and these kinetic
parameters may be applicable to the other G protein in this category such as GS.
And this is what, this is what one means by guesstimated parameters.
And sometimes parameters need to be estimated
from indirect measurements such as time course.
And these can be quite accurate, although you might not get
a parameter that is directly associated with one component or another.
For instance, if there is a time course of ras activation
one can accurately estimate the relative activities of the get and
the gap, but one may not be able to precisely mea, estimate
the kinetic parameters associated with that get or a gap alone.
So there are curve fitting programs that's COPASI allows us one to estimate COPASI
that allows one to estimate these types of programs, these types of parameters.
So your models are only as real as your kinetic parameters, and this is sort of a
very, very fundamental sort of concept that one needs to keep in place.
So, there are some sort of rules we need to follow when building models.
These are rules that we follow in my own lab all the
time, and I would encourage other people who build these models to
follow these rules so that your models are likely to be realistic
or reasonably realistic representation of the systems that you wish to study.
So the first rule is do not oversimplify the model.
And as I told you previously, when
you require different isoforms of the receptors.
Identify the isoforms and use them as di-, distinct entities, and compute your model.
You, incorporating these levels of details.
One needs to build models with enough detail
to provide these non-intuitive hypotheses that arise from simulations.
So this is sort of dealing with details is very important.
And so this is a cautionary tale for students to
come from say applied math or computer science or other back,
or engineering background and say, oh, I know how to build
a model, so I can do it without knowing any biology.
In reality, you cannot.
So you need to understand some level of the biological some level of detail of
the biological system and make choices of how much detail
the model needs to sort of address the questions that you might be interested in.
Use available experimental parameters to data
to obtain realistic, and reasonable parameters.
And so this is sort of very, very
important that your parameters are realistic, they are in
the real range, so if a protein is present
at at, say, 10 nanomolar concentration in a cell.
If you assume the protein to have a concentration of one
micromolar or ten micromolar, you will be so off in your
reaction concentrations that the prior, that the model will compute
stuff that are unlikely to represent anything real happening with in the cell.
So one needs to get reasonably realistic parameters to compute these models.
Once you set the entrance, once you set
the parameters based on what is experimentally known,
it is really, really important not to tweak
or change the models to show a desired behavior.
Personally, I think the worse way to do a model is to say, I would like to observe a
certain behavior and then sort of change the parameters til you see the behavior.
When one does this, what happens is that one loses
any sort of anchoring into the realistic constraints the system
has, and so the behavior may be observed in the
model but may be never observable in real life experimentally.
And this sort of ability for the model to
do anything is what is typically called a spherical cow.
And here I have a picture of the spherical
cow taken from [LAUGH] an Italian website on the web,
and you can see that depending on how many
parameters you put, you can do anything to the models.
So sort of the pejorative way of stating this
is that give me nine parameters and I will build
you a spherical cow, and give me a ten to one and I will make the cow back its tail.
So you really don't want to, your models to be the proverbial spherical cow.
So if you want to know what your system is capable of displaying a behavior or not,
start with the parameters, that you estimated in an unbiased fashion from
the experiment, and then conduct a systematic
parameter variations, and then do, and then as
you vary these parameters, you can see
whether you're in a realistic range or not.
So for instance if you start at a certain set of parameters that
you estimated a two fold to ten fold variation may be realistic, in
most cases and so the behavior changes a lot and you might presume
that you might see these behaviors not very often, but at least sometimes.
But if you need to change vary the parameter by
a factor of 100 to 1,000, this is unlikely to happen
in real life, and so behaviors that require extreme changes in
parameters are not likely to be observable in experimental assistance.
So, once one settles on these parameters, and sort
of stays in a reasonable realistic range, the issue is
how does one solve the systems of ordinary differential equations,
to obtain sort of the results one needs to see.
The, the common or the classical way of solving ODE's
is to use what is called the forward Euler integration.
And I really can't, I don't have time or sort
of, can't go into the details of how this is done.
One needs to take a, a differential equation class to do this.
And it may not be necessary for most
biologists, because there are programs to read such as
Mat-Lab and Octave, and others where you don't
need to know how a differential equation is integrated.
You just have to know what is useful for your kind of model to use.
So the forward Euler sort of
integrate from one point to another by moving
a short distance along a tangent to this curve.
What is really important to know, even without the details, is the
accuracy of the method depends on how short the distance or the time staff is.
Euler's matters are historically important, but
they're currently not used that often
as they are neither versatile, or they're neither very accurate or versatile.
But, improvements of the Euler methods such as the
Runge-Kutta method, which is most commonly used to solve ODEs,
and most math lab solvers, several math lab solvers,
use these Runge-Kutta type methods to solve systems of ODEs.
So, solving PDEs is a little bit harder.
Analytically, most often, PDEs are solved by recasting them as ODEs
using a method called the separation of variables and then solving the ODEs.
But most often in biology we don't really go for analytical solution or it's not
possible, and so the two commonly used numerical approaches for solving PDEs.
Either the finite element method or the finite volume method.
The finite element method
in the finite element method, we, the continuous domain is divided into smaller
parts called elements by enforcing a mesh onto it, and this allows us to sort of get
a system of equations that governs the
flow of entities such as proteins between these,
discrete elements, and then we nu, nu,
numerically solve the system of equations that govern
both the reactions and the flow of entities between these elements.
In the finite volume method, we use, also use a mesh of a defined size.
The volume refers to the volume surrounding a point on the mesh.
here, typically the surface integrals are
converted into volume integrals and solved numerically.
This method is most suitable for cell biological models with diffusion
and is the one which is used in the virtual cell.
There are a number of software suites that are available for numerical computations.
Among the widely, most widely used one is a program, Matlab which is used for a
variety of numerical computation including ODE and PDE models.
We teach all our courses using Matlab, and since the
company is not going to provide Matlab for free, for the
Coursera course during period for the Coursera courses, the dynamic
of modeling course taught by [INAUDIBLE] will also use Matlab.
Another widely used program is Mathematica, which is also
a commercial software for a variety of numerical competition.
There are some free software that are available as well.
GNU Octave is free software suite which is compatible with Matlab,
and is often used by people who do not want to purchase a license for Matlab.
Virtual Cell is another free modeling and, and analysis
software that has unique capability for partial differential equation
models in conjunction with imaging experiments and we often
use Virtual Cell for our spatial models in my lab.
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