March 30

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


 * Chemical species : Xi
 * Either different molecule (repressors one and two)
 * Same molecule different conformation (different conformation of DNA)
 * Same molecule in a different location


 * Reaction routes
 * Some reactions are a function of I (flipping a bit)
 * Rule : bit flips when reaction takes place


 * State
 * Given by the array of species
 * Can't measure every concentration
 * Interrogate a subset of the system


 * 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


 * 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


 * In the time interval, Tu, there is a probability of changing from Xo to Xi
 * P(Tc>t) = e^(-Iu*t)
 * Requirment : Tu>Tc


 * What can go wrong:
 * Remain at Xo, thus Tc t)=e^(-Iu*t)
 * Key points
 * Iu needs be fast enough to allow TcTu
 * Tci=T*, want 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?

 * 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 ...


 * 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
'''**PCR with respect to time
 * Bulk measurements for timescale of flipping
 * Plate reader
 * 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