HC 10 - Systems Modelling Flashcards
hoorcollege 10
Moleculaire interacties (celfunctie) en regulatie hangt af van … interactions van duizenden macromoleculen
transient
For complete picture, what is needed to judge importance of a situation?
Quantitative information
-Concentrations
-Affinities
-Kinetic behavior
For molecular interactions, what do we need to describe?
-Molecule interaction
-Catalyse reactions
-Change of molecules in time
-Gene Y promotes gene Z for example
Types of interaction
-Inhibition of geneX to geneZ > transcription repressor X inhibits expression if Z
-Promotion via activator X
Gene transcription regulation
-RNAp(olymerase) binding site within promotor
-TFs with X binding sites within the promotor
> binding to DNA
> expression gene Y
> increased transcription
The time for transcription and translation is
somewhat equal
Gene transcription regulation: activator characteristics
Activator increases rate of mRNA transcriptions when bound to promotor, it typically transits rapidly between active and inactive forms (e.g. phosphorylation state)
Repressor mechanism of action
Blocks RNAp for binding by binding the promotor
Interaction model between protein (TF) A and promotor p-x
A + p-x <=> A:p-x
kon and koff are the rate constants
Rate of complex formation and dissociation for promotor and protein
complex formation: d[A:px]/dt = kon[A][px]
complex dissociation: d[px]/dt = koff[A:px]
what is kon
Number of productive collisions per unit time per protein at a given concentration of px
[kon] and [koff]
[kon] = M^-1s^-1
[koff] = s^-1
these are different because the units on both sides of the equations must be equal.
At steady-state, the promotor model looks like
Rate of formation = rate of dissociation
kon[A][px] = koff[A:px]
[A:px] = (kon/koff) [A][px]
=K[A][px]
What does K mean in the promotor model? And K_D
K = kon/koff: association constant in M^-1
K_D = 1/D: dissociation constant in M
When there is a strong interaction between A and px, K becomes larger/smaller
larger
In general we know the amount of total px: pxT. Substitute to a equation of [A:px] at steady state
[px] = [pxT] - [A:px]
[A:px]steadystate = K[A][px]
[A:px] = (K[A]/(1+K[A]))*[pxT]
The production rate of protein X is determined by … in the promotor model
the occupancy of the promotor
bound fraction of promotor
[A:px]/[pxT] = K[A]/(1+K[A])
Transcription rate
beta * K[A]/(1+K[A])
beta: represents binding of RNAp and the steps to mRNA > transcription rate when activator is bound
Protein production rate
beta * m * K[A]/(1+K[A])
m: rate of protein made per mRNA
Protein degradation
-Degradation leads to exponential decline in protein levels
-Mean life time tau
-takes active degradation and dilution due to cell growth into account
Protein degradation rate
= [X] / tauX
concentration / life time (halfwaardetijd)
A low tau means a … degradation rate
high
d[X]/dt formula
= protein production rate - protein degradation rate
= beta m (K[A]/(1+K[A])) - ([X]/tauX)
[Xst] formula (at steadystate)
[Xst] =beta m (K[A]/(1+K[A])) tauX
> fill in zero for d[X]/dt
The faster X is degraded (smaller tau), the … time is needed to reach steady-state
less
Faster response means …
higher metabolic cost
> proteins are produced and degraded at higher rates
Solution of differential equation
X = [Xst] (1-e^-t/tauX)
by integrating
When using a repressor model, we are interested in the unbound fraction. The formula is:
px = pxT-[R:px] and [R:px] = K[R][px] give
[R:px] / pxT = K[R]/1+K[R]
unbound factor = 1 - bound factor
= 1 - K[R]/(1+K[R])
= 1/(1+K[R])
At 0.5 bound factor …
1/K_A and 1/K_R
Protein production rate, d[X]/dt and [Xst] based on repressor model
ppr = beta * m * 1/(1+K[R])
d[X]/dt = beta * m * 1/(1+K[R]) - [X]/tauX
[Xst] = beta * m * 1/(1+K[R]) * tauX
Are the beta, m, and tauX functions different for the activator and repressor models?
No
Solutions of
dx/dt = -x
dy/dt = -2y
x(t) = e^-t
y(t) = e^-2t
because the x’(t) is -1 * e^-t which is -x
What is the steady state of
dx/dt = -x
dy/dt = -2y
0 = -x
x= 0
0 = -2y
y = 0
steady-state at (0,0)
Initial condition
Initial values for x and y (starting point of model)
write as a vector
dx/dt = -x
dy/dt = -2y
(-x)
(-2y)
vector format
(dx/dt)
(dy/dt)
> direction from starting point (change)
Why use vectors in model?
Solutions of differential equations are shown in a field over time with multiple paths dependent on initial values
to fill in the vector (-x | -2y), you need
the starting point (x,y)
In the direction field, what is shown
Directions of x and y based on different starting positions and their differential equations
> moving to steady-state(s)
What is a trajectory
A path of x and y in the direction field from a given starting point, by following the direction vectors
> shows the change of x and y over time
What to do with negative time in a time plot (plot x and y as two lines over time > solution of differential equation)?
Neglect it, does not exist in biology
From the directory field, you can … the solutions of the trajectory differential equations
sketch, by following the line over time and checking the development of the x and y values
What is a nullcline
The lines/conditions in which x or y do not change
> the steady-state is the condition in which both x an y do not change
>dx/dt or dy/dt = 0
How to calculate nullclines
dx/dt = 0 for x nullcline(s)
dy/dt = 0 for y nullcline(s)
How to calculate steady-state
Calculate nullclines and fill in the x and y values for the dx or dy/dt =0 in the other formula and match the x and y values for steady-state(s)
Where is the steady-state found in the directional field?
Crossing point of nullclines
At a x-nullcline, x cannot change. How does the trajectory change in x while crossing the x-nullcline
From this point, it changes in y and from that point it can change in x again
A swirl in the trajectory means
oscillations
How to draw direction field
Take some random starting points and fill in differential equations to calculate some vectors
If y approaches a steady-state, but X reaches till infinity and won’t, than there is …
no absolute steady-state, this is realistic in a cellular system
Instable steady-state
There is just one steady-state but the starting point has to be that value, or it will never reach it and trajectories will reach a stable steady-state or reach till infinity
> dx/dt = x + 2y and dy/dt = -y
>y-null: y=0
>x-null: y=-0.5x
> steady-state instable at (0,0)
Positive feedback: promotors can have multiple adjacent binding sites for the same TF. Regulators can interact: more expression than normal > give the formulas for bound fraction for activator or repressor
bound fraction
= (K_a[A])^h / 1+(K_a_[A])^h
or
=(K_r[R])^h / 1+(K_r_[R])^h
What is the Hill coefficient h
degree of cooperativity
Increasing h means
dependence of binding on protein concentration becomes steeper
Hill functions are
Sigmoid functions
Positive feedback can give a system …
switch-like response and bistability
bistable system provides system with a ..
memory > present state depends on its history: hysteresis
Bistability steady-states
-One instable steady-state
-Two stable steady-states
Protein X inhibits Gene Y and Protein Y inhibits gene X. Give the differential equations
dX/dt = Beta_X * m_X * (1/1+(K_Y_(Y)^h_Y) - [X] / tauX
dY/dt = Beta_Y * m_Y * (1/1+(K_X_(X)^h_X) - [Y] / tauY
> because: X is the repressor of Y
x-nullcline for positive feedback through inhibition of X and Y
Fill in 0
X = tau_X * beta_X * m_X * (1/1+(K_Y_(Y)^h_Y)
> y-nullcline
tau_Y * beta_Y * m_Y * (1/1+(K_X_(X)^h_X)
When is a steady-state instable when looking at the direction field
If the trajectories are not attracted by this steady-state
Region of attraction: If the region of attraction around one steady-state is large, then …
If the region of attraction around one steady-state is large then most cells in the population will assume this particular state
The state is likely to be … by … cells
inherited by daughter cells (same steady-state)
> minor perturbation due to asymmetric distribution of molecules in cell division, but rarely sufficient to switch from one steady-state to another
Positive feedback coupled to cooperativity will oftern be associated with systems requiring …
stable cell memory
[Xst] is the … and [Yst] is the
nullclines
Nullcline analysis
Use nullclines to calculate steady-states
