Introduction

We’ve established in a very early stage of the design process that we were going to use DNA methylation as the molecular mechanism to create our storage mechanism. DNA methylation, usually part of the Type II restriction system, is performed by a group of proteins called the methyltransferases. In choosing DNA-methylation, we were heavily inspired by the DamID technology which has been originated by the Bas van Steensel . By fusing a bacterial methyltransferase (MTase) to a eukaryotic transcription factor which is to be characterized, van Steensel was able to infer the transcription factor binding sites on the eukaryotic genome by reading out where methylation had occurred. We thought we could reverse this idea to create memory unit, at the core constituted by an MTase that would only methylate an especially designated genomic region in reaction to the sensing of a signal. For this, we required an MTase that

is not already present in the chassis organism E.coli, has a binding motif of that is not methylated by any other MTases in the system and for which a restriction enzyme (RE) dependent upon the action of this methyltransferase has been identified.

By finding a way of controlling the activity of the MTase, a genomic region purposed to be methylated, named the bit region, would only be methylated when the cell would be induced to do so by the signal. Finally, the methylation status of the bit regions can then be assessed using digestion with this RE followed by a band length analysis of a gel electrophoresis of the product. This is process is covered more in depth in section {sub:gelinfer}

Choosing a methyltransferase

We wrote a python script to mine the [http://rebase.neb.com/rebase/rebase.html REBASE] database of REs and MTases for a MTase that meets our needs. We found an ideal candidate: M.ScaI, originally from Streptomyces caespitosus. Now, the only remaining design question was how to control the activity of the fusion protein as we would want it to only be active in the presence of a signal. We could either try to control its presence in the cell by induced transcription or try to directly fuse it to a inducible transcriptional regulator, hoping that the blocked activity of the latter in absence of a signal would also prohibit the MTase from methylating the bit. This design question will be more elaborately discussed in section {chapter_molecular}.

Extending the idea to multiple bits

Extending the design idea further, we realized that it should be possible to store the presence of different signals – multiple bits – simultaneously by fusing DNA binding proteins to the methyltransferase and adding their corresponding DNA binding motifs to the bit regions of the DNA. Either traditional bacterial transcriptional regulators could be used for this purpose or the more extendable Zinc-Finger Array (ZFA) technology . The latter allows the construction of highly specific proteins to custom DNA motifs. The large increase in binding affinity of the resulting MTase-ZF fusion protein would have to make sure that the same MTase can be used for different signals with a very low chance of cross-talk between FPs and other signals’ bits.

Inferring the time of signal onset

Numerous copies of identical plasmids are often present in single cells and plasmids replicate independently of the bacterial chromosome . A plasmid copy number (PCN) has been determined for all plasmids in the Parts Registry, which indicates a likely amount of copies of the plasmid to be present in each cell. Unlike eukaryotes, we discovered that DNA methylation patterns are not copied to the new strand during DNA replication in prokaryotes. This will have to lead to a dilution of the amount of plasmids with a written bit over time, mostly due to cell replication and the ensuing binomial division of the plasmids in the parent cell among the two daughter cells. This seemed like a downside to using a MTase system at first, but it quickly opened our eyes to a very exciting application of this system. Assuming nearly constant proliferation & replication rates for the cells and the within the plasmids, this dilution can be used to infer how long ago the signal of interest was registered by the cell.

Using the here presented model, we will examine how this phenomenon can exactly be used to infer the registration time of the signal by the cell. Another goal of this model is to gain insight into how much the various parameters for growth and degradation speed of plasmid in the model affect the ratio $F(t) = \frac{\text{written plasmids}(t)}{\text{total plasmids}}$. Unknown variables affecting <math>F(t)</math> in a real-life setting would be the time of signal onset, signal duration and signal strength. Additionally the signal response, degradation and proliferation rates of the Cellular Logbook will influence <math>F(t)</math>, but these should be experimentally determinable before actual deployment and application of our system. This way the can be used to measure a signal and estimate its time of onset. This would allow the inferrance of signal onset <math>s_{\text{on}}</math> from the fraction of methylated over total bits in the population. As this model has the sole purpose of clarification of the desired system behaviour, the units in this model are completely arbitrary.

First, let’s model the input signal/compound which is to be registered. Let <math>s_{\text{on}}</math> be the time at which the signal is turned on and <math>s_{\text{off}}</math> be the time at which the signal is turned off. Imagine the to be stationary and positioned along a fluidic stream with the signal to be registered passing the at <math>t = s_{\text{on}}</math> and leaving it at <math>t = s_{\text{off}}</math>. Modelling the signal using the piecewise function <math>S(t)</math> in equation {eqn:signal} seems appropriate. This function has been plotted in Figure {fig:signal}.

<math>S(t) = \left\{ \begin{array}{lr} 0 & 0 \le t \le s_{\text{on}} \\ 1 & s_{\text{on}} \le t \le s_{\text{off}} \\ 0 & s_{\text{off}} \le t \le \infty \end{array} \right. \label{eqn:signal} </math>

File:Signal.png {fig:signal} { Plot of the input signal <math>S(t)</math> with <math>s_{\text{on}}</math> at 3 and <math>s_{{\text{off}}}</math> at 4}

A single cell with multiple identical plasmids on which the gene for the methyltransferase and a so called bit region, which is the region especially purposed to be methylated in presence of a signal, have been placed. In this cell the fraction of methylated plasmids <math>F(t)</math> will then be in direct correspondence to how long ago the signal has been registered if the following plausible rules are true:

• A well stirred cellular system in which no spatial concentrations differences exist and in which all species concentrations are large enough to be approximated by continuous functions. Such a system is describable by a set of ordinary differential equations (ODE’s).
• To easy the analysis of the model, a single unique bit with multiple copies per cell is considered here. The system is easily upgradeable to multiple signals however.
• Assumed is a high response rate ( 3 min) termed <math>\omega</math>, which is the constant rate with which the system responds with methylation to the sensing of a signal.
• We will assume logistic growth for the memory cell population, with a capacity limit of <math>Ca = 5000</math>.
• Alternatively, this maximal population number could also be described as the fraction between the cellular proliferation rate (<math>\beta</math>) and degradation rate (<math>\alpha</math>), as the steady state amount of cells in the population will be determined by <math>\frac{\beta}{\alpha}</math> as a solution to the differential equation <math>P'(t) = \beta P - \alpha P</math>.
• Accumulation of cells due to the division of cells in which te bit has been written is assumed to result in non-written cells; methylation patterns are not copied to the progeny in prokaryotes.
• <math>P_{0}</math> denote cells that have the single bit set to 0, no write event has taken place in these cells.
• <math>P_{1}</math> indicate cells in which the bit has been flipped to 1 in response to encountering the signal compound.
• <math>P_{0+1}</math> indicates the total amount of cells in the sytem, <math>P_{0} + P_{1}</math>

From these rules, the following system of ODE’s has been constructed:

<math>\frac{dP_{0}}{dt} = k\ P_{0+1}\ (1 - \frac{P_{0+1}}{\text{Ca}}) - \omega\ S(t)\ P_{0} - \alpha\ P_{0} \\ \frac{dP_{1}}{dt} = \omega\ S(t)\ P_{0} - \alpha\ P_{1} </math>

<thead> </thead> <tbody> </tbody>

Parameter	Value
k	1.5
<math>\beta</math>	2500
<math>\alpha</math>	.5
Ca	5000
<math>\omega</math>	8

{ Parameter values for the plasmid methylation model} {tab:populationparms}

Using the parameter values of Table {tab:populationparms} a simulation with a duration of 20 time units is shown in Figure {fig:timelapse}.

image { Input signal <math>S(t)</math> with <math>s_{\text{on}}</math> at 3 and <math>s_{{\text{off}}}</math> at 4} {fig:timelapse}

image { Sum of unmethylated and methylated cells, <math>S_{0+1}</math>} {fig:totalcells}

The plasmid population within a is shown to be completely converted to methylated plasmids shortly after . As long as the signal is still present – until <math>s_{\text{off}}</math>, – the bit on all newly copied plasmids will be immediately methylated as the signal is still present. After <math>s_{\text{off}}</math>, the percentage of <math>\frac{\text{methylated} \text{cells}}{\text{total} \text{cells}}</math> will be diluted, due to cell proliferation implicating plasmid division between daughter cells and cell death. The size of the time span with which a hint of methylated cells is still present therefore relates to two factors: positively with the amount of methylated cells at <math>s_{\text{off}}</math> and negatively with the cellular death rate.

The bit writing speed termed <math>\omega</math>, or the rate with which the presence of a signal causes the methylation of a bit, is likely to be very high in comparison to the cellular division rate. The underlying required processes such as the diffusion of the signal across the cellular membrane, the signal inducing gene transcription of the FP, mRNA translation and transcription & and FP diffusion over to the bit all occur well within 5-minute time scales. Every single gene on a plasmid will then be expected to be methylated within at most 5 minutes of registering of the signal.

The assumption that the population will have reached its steady state level (SS), given by the production rate (<math>\beta</math>) divided by the degradation rate (<math>\alpha</math>), before is a plausible one. The Cellular Logbook population will have been stored inside of a vesicle consisting of a non-bacterial permeating membrane with a very small volume. Due to this volume and the rapid bacterial division time (<math>\sim</math> 40 mins), the spatial growth limits of the system will then have been reached in a short amount of time. The steady state-level of the bacterial population has already been reached before the signal is registered by the population.

To reinforce that the following equation always holds in this model; <math>P_{0} + P_{1} = P_{0+1}</math> the following plot is shown, in which the sum of the methylated and unmethylated cells over time is plotted. This clearly shows the limit, specified by the Capacity limit, which is reached around <math>t=6</math>.

Methylated plasmids over time

{sub:gelinfer} The monotonically decreasing value of <math>F(t) = \frac{\text{methylated plasmids}}{\text{total plasmids}}</math> can be used to infer , given that the cellular proliferation rate (<math>\beta</math>) and degradation rate (<math>\alpha</math>) are known and constant and the assuming that all bits are methylated during the presence of the signal. Irrespective of the initial amount of plasmids, the population of plasmids within the single cell will have reached a steady state value of <math>\frac{\beta}{\alpha}</math>. After the signal has leaved the medium in which is present,  will start to decrease as a function of the degradation rate .

<math>\frac{dP_{1}}{dt} = - \alpha\ P_{1} </math>

Integrating this differential equation, <math>P_{1}</math> will be given by: <math>P_{1}(t) = e^{-\alpha t} </math> And multiplying by the steady value <math>\frac{\beta}{\alpha}</math> will yield the amount of methylated plasmids at time <math>t</math>, given that there were <math>\frac{\beta}{\alpha}</math> methylated plasmids at <math>t = 0</math>. <math>P_{1}(t) = \frac{\beta}{\alpha} e^{-\alpha t} </math>

In practice

image { Gel representations for a range of different <math>F(t)</math> values. Complete methylation of all bits results in a single, bright band at the height of the length of the linearized plasmid. Decreasing the amount of methylated bits shifts the intensity of the top band away to the two bottom bands. These indicate the linearized & digested plasmid.} {fig:gelarray}

In a typical laboratory situation, doing a restriction enzyme assay on the miniprep-extracted plasmid DNA out of followed by gel electrophoresis will be the most convenient way to assess the methylation status of the bits. The relative intensities of the gel bands can then be used to infer , which is exemplified in Figure {fig:gelarray}. Unmethylated bits will result in successfully digested DNA fragments and thus two bands of shorter DNA fragments. Methylated bits will not be cut and will therefore result in one longer band, shown more to the top of the gel. Hence the top band and the two bottom ones are mutually exclusive as they indicate the same (linearized) plasmid DNA to either be cut, resulting in the two bottom bands, or not cut, resulting in the top band.

The function for <math>P_{1}(t)</math> is shown in in Figure {fig:readout}. The rate with which the function approaches its limiting value of 0 is directly dependent upon the degradation rate . To get a hands-on feel of the effects on the system an interactive version of this plot, in which , and are controllable via a GUI is included in the attached Mathematica file.

image { Animation of <math>P(t)</math> with its value ranging between 0.01 and 1. Solving for <math>t</math>, the time after which the signal has left/been degraded in the Cell, the solution is shown by the blue dot and the text underneath the plot. The value of <math>P(t)</math> will decrease over time until it becomes 0. An interactive version of this plot has been included in the attached Mathematica file.} {fig:readout}