March 30: Difference between revisions
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****The chance does not decrease with respect to time | ****The chance does not decrease with respect to time | ||
**In the time interval there is a probability of changing from Xo to Xi | **In the time interval, Tu, there is a probability of changing from Xo to Xi | ||
***P(Tc>t) = e^(-Iu*t) | ***P(Tc>t) = e^(-Iu*t) | ||
***Requirment : Tu>Tc | ***Requirment : Tu>Tc | ||
Line 44: | Line 44: | ||
**What can go wrong: | **What can go wrong: | ||
***Remain at Xo, thus Tc<Tu, with Tc driven by reaction rate given by PDF: P(tc > t)=e^(-Iu*t) | ***Remain at Xo, thus Tc<Tu, with Tc driven by reaction rate given by PDF: P(tc > t)=e^(-Iu*t) | ||
***Key | ***Key points | ||
****Iu needs be fast enough to allow Tc<Tu | |||
****Not too fast such that Tci+Tc2<Tu | |||
**P(correct) at first count; Tci<Tu and Tci+Tcii>Tu | |||
***Tci=T*, want T*<Tu | |||
***Tcii+T*>Tu; Tcii>Tu-T* | |||
***Integral ( P(Tci=T*) P(Tcs>Tu-T*) dT | |||
****P(Tci=T*) from PDF = Iu*-IuT* | |||
****P(Tcs>Tu-T*) from PDF = Iu*-Iu(Tu-T*) | |||
**Tl = nothing happens and Tu=pulse; only change state during Tu and Tl can be long, short, unequal, etc | |||
**xi->x(i+1) with rate proportional to Iu given by PDF: Iu*e^-(Iu*t) | |||
**P(X(k*Tu))=Xk=Probability of correct counting at count value K | |||
*Key rules | |||
**tci+tcii ...tck < kTu ; exactly k events over kTu | |||
**tci+ ...tc(k+1) > kTu | |||
**Possion process: probability of k events occur over timescale T = e^(-lamba*T)*(lambda*T)^k.(k!) | |||
***Interval between each event | |||
***Lambda is the rate at which each event occurs: Iu, same as Iu in tu ~ Iu*e^(-Iu*t) | |||
***Time is k*Tu | |||
**Probability that one counts correctly: how will this change if I change Iu, Tu, | |||
*Input signal changes over time | |||
**It peaks up at some point | |||
**During time interval, Tu, there is probability of state change | |||
* Probability of change state tci ~ Iu*e^(-Iu*t) | |||
** Driven species : Iu | |||
** More molecules | |||
* Successful count given by | |||
**tci < Tu | |||
**tcii+tcII>Tu | |||
* I want to have K state changes over total induction time K*Tu | |||
** Implies that intermediate state changes don't matter | |||
*** For example two count within one large induction time and no change within the second | |||
** Probability of this happening is given by Poisson distribution | |||
*** P(k, Iu) = e^(-lamba*T)*(lambda*T)^k.(k!) | |||
Iu is the reaction rate for flipping driven by: | |||
** | |||
**Want k spots within kTu | |||
==What data can we get now?== | ==What data can we get now?== |
Latest revision as of 12:49, 30 March 2009
Chemical counter
- System
- Input : what we want to count
- Variable I that is continuous with respect to time, given by I=f(t)
- Examples: iptg, light, chemical species
- Count peak of I up to some time, T
- Peak : I(t)'=0, I(t)>0 (maximum), above threshold, and minimum between peaks lower than another threshold
- Input : what we want to count
- Chemical species : Xi
- Either different molecule (repressors one and two)
- Same molecule different conformation (different conformation of DNA)
- Same molecule in a different location
- Chemical species : Xi
- Reaction routes
- Some reactions are a function of I (flipping a bit)
- Rule : bit flips when reaction takes place
- Reaction routes
- State
- Given by the array of species
- Can't measure every concentration
- Interrogate a subset of the system
- State
- Predict
- N(t)
- X(t) represents the state of X at time T, which could be Xi, Xii, etc.
- This is a good counter : P(X(t))=X(5)=1, if t=5
- Predict
- Example
- X0 at time=0
- Assume discrete input with T(up) and T(low)
- Rules:
- Do nothing when signal is low
- When high: xi->x(i+1) with rate r and time Tci
- Time given by Tci = time of reaction = exponential distributed with respect to rate = Iu
- Probability that Tci > T; Iu*r^(-Iu*t)
- Probability that Tci > t is given by e^(-Iu*t)
- The chance does not decrease with respect to time
- Example
- In the time interval, Tu, there is a probability of changing from Xo to Xi
- P(Tc>t) = e^(-Iu*t)
- Requirment : Tu>Tc
- In the time interval, Tu, there is a probability of changing from Xo to Xi
- What can go wrong:
- Remain at Xo, thus Tc<Tu, with Tc driven by reaction rate given by PDF: P(tc > t)=e^(-Iu*t)
- Key points
- Iu needs be fast enough to allow Tc<Tu
- Not too fast such that Tci+Tc2<Tu
- What can go wrong:
- P(correct) at first count; Tci<Tu and Tci+Tcii>Tu
- Tci=T*, want T*<Tu
- Tcii+T*>Tu; Tcii>Tu-T*
- Integral ( P(Tci=T*) P(Tcs>Tu-T*) dT
- P(Tci=T*) from PDF = Iu*-IuT*
- P(Tcs>Tu-T*) from PDF = Iu*-Iu(Tu-T*)
- P(correct) at first count; Tci<Tu and Tci+Tcii>Tu
- Tl = nothing happens and Tu=pulse; only change state during Tu and Tl can be long, short, unequal, etc
- xi->x(i+1) with rate proportional to Iu given by PDF: Iu*e^-(Iu*t)
- P(X(k*Tu))=Xk=Probability of correct counting at count value K
- Key rules
- tci+tcii ...tck < kTu ; exactly k events over kTu
- tci+ ...tc(k+1) > kTu
- Possion process: probability of k events occur over timescale T = e^(-lamba*T)*(lambda*T)^k.(k!)
- Interval between each event
- Lambda is the rate at which each event occurs: Iu, same as Iu in tu ~ Iu*e^(-Iu*t)
- Time is k*Tu
- Probability that one counts correctly: how will this change if I change Iu, Tu,
- Input signal changes over time
- It peaks up at some point
- During time interval, Tu, there is probability of state change
- Probability of change state tci ~ Iu*e^(-Iu*t)
- Driven species : Iu
- More molecules
- Successful count given by
- tci < Tu
- tcii+tcII>Tu
- I want to have K state changes over total induction time K*Tu
- Implies that intermediate state changes don't matter
- For example two count within one large induction time and no change within the second
- Probability of this happening is given by Poisson distribution
- P(k, Iu) = e^(-lamba*T)*(lambda*T)^k.(k!)
- Implies that intermediate state changes don't matter
Iu is the reaction rate for flipping driven by:
- Want k spots within kTu
What data can we get now?
- We can induce and get uni-directional flip
- We can measure growth in fluorescent signal
What are the critical issues
- How fast does the flippee flip?
- Current data lumps all events, so we need to dis-aggregate
How do we de-couple the events?
- Big question: what is the rate limiting factor in the process?
- When is time between induction and binding?
- Localization assay with flipper-GFP fusion
- When is time between induction and flipping?
- PCR with respect to time
- Bulk assay, but don't know distribution
- How do we do this quantitatively?
- Now, we amplify spontaneous flipping ...
- PCR with respect to time
- How fast does it take to visualize GFP?
- Measure signal with respect to time following induction under control of same promoter
- What drives the time of flipping?
- Inducer length
- Int / Xis expression dynamics
Modeling
- Flippee
- System
General experimental tools
- Reporter
- Gemini
Measure Int / Xis leakiness
- Put reporter behind the Int or Xis promoter
- Measure signal when not induced
- Tag the Int with fusion
- Direct measurement of the flipper
mRNA quantification
- mRNA coding for flipper (int)
- Tag mRNA with florescence to quantify
Measurement of DNA binding
- FP-Int fusion
- Low copy Int expression and visualize signal only when bound
Recombination
- Bulk measurements for timescale of flipping
- Plate reader
**PCR with respect to time
- Induce
- Stop reaction
- PCR
- How sensitive is the PCR single with respect to the number of flipped templates?
- Single cell measurements for timescale of flipping
- See variability across the population
- Get the distribution
- FP on plasmid measurement
- Distance between two signal cases is ~100-80nm on plasmid
- Plasmid replicates and is not synchronized with the cell
- Therefore two dots could come from