lectures

degradation

basic reproduction number — number of people infected by an individual while they’re infectious

NOT on n, but on t — exponential growth

no matter how big n is, Tdouble is the same

number of immune people which stops the infection spread

measured kinetics of enzymatic reactions, did initial rate measurements

S+E = C = E + P

in their model the substrate is in surplus and slowly gets converted into product

measured oxygen partial pressure in the air

the difference of Hb (haemoglobin) loading with O2 increases

the data can be fitted with a very simple mathematical expression:

he only found out that Hb has 4 independent O2 binding sites, each with affinity K

no clue about cooperativity

Adair

provided an excellent fit for the data:

O2 bidning to one of the hemes increases affinity if other hemes

Punling in 1935

obviously wrong

in 1959 by Perun, he discovered the first Hb structure, proved that hemes are far from each other

Monod Wyman and Changeux in 1965

1) Hb subunits are identical

2) each subunit can exist in 4 confirmations : tense and relaxed

3) the Hb tetramer can exist in either all-tense or all-relaxed confirmation — the two states are in equilibtrium

4) both tense and relaxed conformations bind O2 just with different affinities

L = T/R

y = number of bound states / total concentration of binding sites

high affinity - low dissosiation constant Kr (Kon/Koff) and high Kt (Koff/Kon)

protein translation, DNA replication, DNA repair, immune system

error rate = rate of incorporation of a wrong AA / rate of incorporation of a correct AA

energy consumption in the process

it allows the cell to encode memory of past events

dependence of the final state on the initial condition (or the system in general)

metabolic energy input - ATP or GDP

ability of a system or circuit to exist in two distinct stable states

when x is intermediate (CDK), the system will have 3 steady states, two of which will be stable

stable - attract the system’s dynamics for a long time

unstable - serve as boundaries for dynamics

positive feedback & ultrasensitivity

by linearisation

eigenvalues of the Jacobian evaluated at the fixed point

for fixed point to be stable, both EV must be negative, have a negative real part

n > 1

in the Hill’s equation

this way the system can become multistable

when the time derivative vanishes, dx/dt = 0; and the degradation rate is equal to production rate

when the system near a fixed point mover towards this fixed point, from any direction - stable FP

otherwise - unstable FP

over time it will converge to one of the stable pointd

a quantitative change in the behaviour of the system (parameter) leads to a drastic change in the system — like, switching from bistable to monostable

it supports multiple phenotypically-distinct cell states

and enables irreversible responses to transient input signals (hysteresis)

Consider the postive autoregulation circuit in its bistable regime, with xx at the upper stable fixed point. Now, suppose some transient perturbation occurs. Perhaps the value of γγ is suddenly increased because the cell has begun dividing at a much faster rate. If this perturbation is large enough to cross the bifurcation boundary, then the system becomes monostable, and xx will be driven to the only stable fixed point, at a value closer to what was previously the lower fixed point in the bistable regime. Once the perturbation ends, γγ returns to its original value, and the system regains bistability. But, if the perturbed value of xx lies within the basin of attraction of the lower bistable state, then the system will remain in the lower state. In this scenario, a transient perturbation will have led to a permanent state switch. In this sense the system has a “memory” of the transient event.

locally approximate a nonlinear dynamical system by its Taylor series to first order near the fixed point & examine the behaviour of the resulting simpler linear system

if all real EV < 0 the fixed point is linearly stable

if all real EV > 0 the fixed point is linearly unstable

if imaginary EV != 0 for unstable fixed point, the trajectories spiral out, potentially leading to oscillatory dynamics

if real EV = 0 for one or more EV, with the rest of EV < 0, the fixed point lies at a bifurcation

for f(x) we consider u = x - x*

dx/dt = du/dt = f(x) + J * vector u

du/dt = J*u

u(t) = c * e^ (lambda*t)

c - some vector, lambda - some constant

if lambda > 0, we will have exponential growth in this point, so the point will be considered unstable

BUT if lambda < 0, the system will condense in this point

saddle-node (fold)

transcritical

pitchfork (super- and subcritical)

value of dx/dt == in other words a change of function

at p=pc two fixed points merge

example: ultrasensitive auto-activator

stable & unstable FP collide at p=pc and exchange stabilities

example: positive feedback regulation

devise a strategy to maximise reward

(chess games, self-driving cars)

global minimum is hard to find

data: sampled from a different distribution

overfitting: flexible model, few noisy training data

biased training data, not representative for validation data

lack of statistical associations

umbrella term for techniques to prevent overfitting of models while allowing sufficient flexibility to avoit overfitting

Few model parameters

At least 10 times more training data than model parameters

diffusion - movement of molecules along the concentration gradient

Fick’s law: j = -D*(delta c / delta t)

delta - partial der in this case

drift - directed motion through molecular motors j = v * c

velocity v (usually due to a force)

intial condition

c(x, t = 0)

and

boundary condition (what happend at the end of the spatial domain)

nothing goes beyond it -> the flux outside the boundary is 0, gradient is 0 too

normal distribution

r = sqrt(2sDt)

gradients established by a localised source depend on both diffusion and degradation of the morphogen

sqrt(D/delta)

what’s the probability of finding n molecules in a cell?

presenting data and predicting outcomes that account for certain levels of unpredictability or randomness

when we know parameters & initial condition, solution will be determined

in the state Pn we can generate or lose 1 molecule of mRNA, so the dPn/dt = probability of loosing this state - probability of coming to this state

1/time

analytical solution: P~e^At

in SOOOME cases we can get a distribution out of it (a steady state distribution)

but we can solve it numerically by assuming an upper bound for n

or simulate stochastic realisation (gillespie algorithm)

or generate mean, variance, skewness of Pn(t) and derive ODEs for them via moments of the distribution

plug “n” into the master equation

n = 0, n = 1, n = 2 usw

detailed balance - system’s ss can evolve, the cyclic fluxes transition rates are 0 between any pair of states

the Pn for all n follows a Taylor’s series of e^v/delta and still includes Po

steady state distribution of mRNA molecules follows a poisson distribution

• for large n looks like normal

• mean = variation = lambda

• relative deviation decreases with number of molecules

• coefficient of variation is independent of how data is measured = st. dev / mean

• range of values

p is a non-negative real number

• normalisation

degree if belief 1 is assigned to outcomes that are certain

• additivity

the degree of belief that one of several alternatives is true shall be the sum over alternatives pi or pj or pk = pi + pj + pk

joint: p(A and B) = sum of A and B p(i) / p(all i)

conditional: p(A|B) = sum of A and B p(i) / A true p(i) = p(A and B) / p(A)

p(A and B) = p(A|B)p(B) = p(A)p(B)

the quantile ar probability of q is the value of xq so that x is below xq with probability q

• probabilities are degrees of belief

• we can assign probabilities to anything, including non-repeatable events & parameter values

• probabilities are correct if they reflect the knowledge of a rational unbiased observer

• probabilities are limits of frequencies after many repeats

• only reproducible experimental outcomes have probabilities

probabilities are correct if many repetitions of an experiment give h~p

h - frequency

the p of measurement to give data D under the parameter a and background knowledge

p(D|a,B)

observational data

it quantifies the strength of support the observed data lends to posible values for the unknown parameters

a -> p(D|a,B)

p(D|B)

p(a|D,B) = p(a|B)p(D|a,B) / p(D|B) ~~~ p(a|B)p(D|a,B)

or

p(A|B) = p(A^B) / p(B) = p(B|A)p(A) / p(B) ~~~ p(B|A)p(A)

where p(A|B) is prior

p(B|A) probability of data gived the model parameter

p(A) likelihood

p(B) normalising factor, can be dropped

background+observational data

can be used to make predictions about future events

aims at learning the characteristics of a population from sample: parameter estimation, hypothesis testing, confidence interval construction, presiction

way of learning and determining the population’s parameter based on model fitted to the data

determining the likelihood of observing this sample data under different assumptions

aka which hypothesis is more consistent with observed data

• when new inpedendent experiments can ge incorporated in any order at once

• Bayes rule follows immediately from the definition of a conditional probability

• parameter distributions as states of knowledge must be updated in this way to be consistent

split the data and see if one half predicts the other

or

re-generate predicted mRNA counts

large prior -> more flexible model

models dynamics described by a master equation

DM overlook a lot or microscopic interactions and details

way to go: do a stochastic “discrete event” model instead

• individual reactions are modeled separately as discrete events happening in time

• for a reaction to happen, reactants need to bump into each other in a certain correct way

• so we can model probabilities that these bumps will occur

mathematical function which describe either possible outcomes of a sample space or range of the measurable space

eg: sample space (heads/tails) RV (+1/-1) probability (1/2)

when model gives good predictions for training data but not for new data

it happens when model can’t generalise well and fits too closely to training data, for example.

optimisation algorithm that finds local minima of differebt functions

gradient - slope of function. the higher it is, the faster will the model learn. zero slope - model stops learning.

supervised ML algorithm that is good for binary classification problems (sigmoidal curve)