[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 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.

Mathematical Representations of Cell Biological Systems I

Introduction to Systems Biology

Conheça os instrutores