Model
To control cell cycle, we designed a genetic circuit to control replication by blocking the replication initiation
site, oriC. But it brings us three questions:
A. What would happen if we block the chromosome replication?
B. Can we mathematically describe this abnormal process?
C. Is this approach effective enough to control cell cycle?
After Dry Lab’s analysis, we found that blocking oriC is well enough to control cell cycle, and even better since
the initiation nearly determines the cell cycle.
How to observe replication procession in real-time?
Limited by experimental apparatus, we couldn’t observe the replication procession in single cell directly. So Dry
Lab have to build a model based on a series of previously published rules to describe the coordination among
chromosome replication, cell growth and division.
So here comes three main goals of modeling this year:
A. Improving model to mathematical formula.
B. Solving the paradox, that a cell can’t grow in exponential rate after blocking replication, emerged during simulation
by defining a new parameter “d” to supply it.
C. Inferring the internal replication procession only by volume.
In the process of modeling, we encountered many challenges, such as bacteria initiates multi-replication process
during one cycle. According to reference, it is very difficult to be mathematically formulated, but with some
smart trick we can formulate it into a mathematical function.
Figure 1. The different stages of synchronization. Ton: the light triggering the synchronization system is on. And
CRISPER system will block oriC. Tr1: It won’t exert any effect when oriC is not blocked since it can’t initiate
new replication fork. Ts：time of becoming single chromosome.
Figure 2. The different stages of the bacteria returning to normal. Toff: the time it takes from turning off the
light to oriC is unblocked. Tr2: time it takes for the bacteria returning to the normal state since Oric is unblocked.
Section A: volume and replication model
There’s a lot of attributes for a single bacterium such as volume, weight, concentration of different kinds of proteins
and RNA, the amount of DNA and etc., but the accurate concentration of these things can’t be directly detected
due to limited experimental condition. This is a huge obstacle for the modeling’s accuracy. After discussion,
we find that though the concentration of dCas9 and gRNA of the single bacteria can’t be directly acquired, the
volume of cell can be the easiest and most accurate parameter we can get. So that our project can be evaluated
on single cell level other than a whole bunch of colony just because we took the advantage of the volume of the
bacteria which is totally measurable and reliable.
Thus we improved our previous work, made it formulized and capable to be put into software. We read some paper in
the relevant research field and summarized some rules of bacterial replication growth and division, and applied
them in our project. The rules are as following:
Bacterial rule of growth:
$V_{new} =V_{old}e^{\mu T}$
Bacterial rule of triggering replication initiation:
$V>2^{N}V_{std}$
Bacteria will divide after C phase and D phase:
$T_{C}+T_{D}$
Figure 3. Diagrammatic sketch of the coordination among chromosome replication, cell growth and division
Section B: inferring the progress of replication inner the bacteria
1. Take a bacteria’s photo to get the volume Vold, Wait for the twait and take one photo of the same cell to get
the volume Vnew
2. Calculate the growth rate according to the formula.
$\mu =\dfrac {nV_{nen}-lnV_{old}} {t_{wait}}$

3. It can be inferred that the upper limit of the replication folk Nmax is $\lceil log_{2}V_{std}e^{\mu(T_{C}+T_{D})} \rceil$ when the volume of bacteria is V.

4. The volume when the bacteria complete stage C and getting into stage D: $V_{D} =V_{std}e^{\mu T}$ 5. the progress of the smallest replication folk $$x= \begin{cases} \dfrac {lnV-lnV_{std}} {\mu T_{c}}- \lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu T_{C}} ,V>V_{std}\\ \\ 0,V < V_{std} \end{cases} $$ Note: x is greater than 100% means the bac complete stage C and reached stage D. 6. Replication process of the replication forks $$\begin{cases} \quad \ \ \ x\\ \\ \dfrac {1} {log_{2}e^{\mu T_{c}}} + x\\ \\ \dfrac {2} {log_{2}e^{\mu T_{c}}} + x\\ \\ \quad \ \ \ .\\ \quad \ \ \ .\\ \quad \ \ \ .\\ \\ \dfrac {N_(max)} {log_{2}e^{\mu T_{c}}} + x \end{cases} $$ 7. Inferring how long will the bacteria divide: $T_{C}+T_{D} - \dfrac {lnV-lnV_{std}}{\mu}$

8. Inferring how long will it take for the bacteria to form the haploid chromosome if all the Oric is blocked. $T_{C}+T_{D}-xT_{C}$

9. The range of periodically changed volume $[\dfrac {V_{std}e^{\mu(T_{C}+T_{D})}}{2},V_{std}e^{\mu (T_{C}+T_{D})}]$ 10. The interval between division under the current environment. $\dfrac {ln2}{\mu}$ Section C: when will the normal cells be actually affected after blocking oriC When the normal bacteria are going to form a new replication fork and oriC is blocked at that time, the replication process is actually inhibited and normal cells become abnormal. $T_{r1}=\dfrac {ln2}{\mu}-xT_{C}. $ Equal with $T_{r1}=\dfrac {ln2}{\mu}-(\dfrac {lnV-lnV_{std}} {\mu}-\lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu})$ Section D: when will the abnormal cells recover to the normal stage. According to the state of bacteria under the effect of CRISPR/dCas9, cell volume will become much larger than to normal one, but it only have one chromosome. If we allow it initiates new replication process (release its OriC), it would initiate many OriC to recovery itself, but the previous model didn’t describe this phenomenon, so we define a new parameter “d” to make supplement. ASSUMPTION: Bacteria’s rule of growth: $V_{new} =V_{old}e^{\mu T}$

Bacteria’s rule of triggering replication innitition: $V>2^{N}V_{std}$

Bacteria will divide after phase C and phase Dcomplete, $T_{C}+T_{D}$ The trigger condition to initiate replication under abnormal situation: the shortest interval of forming two replication forks: d. if oriC is blocked within Tr1, the bacteria is not effected since the inhibition of Oric hasn’t actually change the state of bacteria. If oriC is blocked longer than Tr1, we have several analysis: The constraint condition of recovering to normal: $\dfrac{lnV-lnV_{std}}{\mu}-N\dfrac{ln2}{\mu}-k(\dfrac{ln2}{\mu}-d) <0$ K means the number of replication forks, d means minimum folk formation interval, N means the number of replication forks exist in the bacteria. The above formula is transformed, so that it can be solved directly: $k>\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}$ Because k is an integer, the formula add. so it’s better for computer calculation $k=\lceil\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}\rceil$ We get the mean of k, the dividing time for the bacteria to recover to normal is: N+k Finally calculate the time for bacteria to recover from abnormal to normal after oriC is released. $T_{r2}=T_{C}+T_{D}+d(k-1)$ It is worth notice that when the number of bacterial liabilities is less than 2, the minimum interval between two bifurcation must be less than ln2 / u, otherwise the bacteria will be dragged down by the new debt on the way of repayment and will not return to normal. Section E: evidence from experiment aTc has being added for 143min. At this time, the bacteria has passed through the Ton period in the model and aTc has already affected replication initiation. So an ultralong bacteria occurred. The temperature will be heated to 45℃ after 85min After being heated for 55min, the longest bacteria in the field of view is under the Tr2 stage. (At this time, there is no aTc in LB medium in the Microfluidic chip).this bacteria is recovering. The initiate stage is remained in the C period after dCas9 is released. After being heated for 133min, the ultralong bacteria has divided many times and recovered to the normal stage. Dividing many times in a short time also proved the validity of the model in a certain degree, which means the attribute “d” exists. We have realized this point before doing Microfluidic experiments through the model, and built mathematical model above. Approximate and preliminary measurement result of "d": 18 minutes. Future work: real-time fitting strategy Model integrates into software. Software controls hardware. Hardware regulates optogenetic system. Optogenetic system regulates growth rate. Growth rate affects the shape of bacteria. The shape of bacteria is analyzed by model. References 1. Stephen Cooper (2006). Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research. Theoretical Biology and Medical Modelling, 3:10 2. M Wallden, D Fange, EG Lundius, Ö Baltekin, J Elf (2016). The Synchronization of Replication and Division Cycles in Individual E.coli Cells. Cell, 166(3):729-739. 3. Cooper S, Helmstetter CE (1968). Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol 31(3):519–540.

3. It can be inferred that the upper limit of the replication folk Nmax is $\lceil log_{2}V_{std}e^{\mu(T_{C}+T_{D})} \rceil$ when the volume of bacteria is V.

4. The volume when the bacteria complete stage C and getting into stage D: $V_{D} =V_{std}e^{\mu T}$ 5. the progress of the smallest replication folk $$x= \begin{cases} \dfrac {lnV-lnV_{std}} {\mu T_{c}}- \lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu T_{C}} ,V>V_{std}\\ \\ 0,V < V_{std} \end{cases} $$ Note: x is greater than 100% means the bac complete stage C and reached stage D. 6. Replication process of the replication forks $$\begin{cases} \quad \ \ \ x\\ \\ \dfrac {1} {log_{2}e^{\mu T_{c}}} + x\\ \\ \dfrac {2} {log_{2}e^{\mu T_{c}}} + x\\ \\ \quad \ \ \ .\\ \quad \ \ \ .\\ \quad \ \ \ .\\ \\ \dfrac {N_(max)} {log_{2}e^{\mu T_{c}}} + x \end{cases} $$ 7. Inferring how long will the bacteria divide: $T_{C}+T_{D} - \dfrac {lnV-lnV_{std}}{\mu}$

8. Inferring how long will it take for the bacteria to form the haploid chromosome if all the Oric is blocked. $T_{C}+T_{D}-xT_{C}$

9. The range of periodically changed volume $[\dfrac {V_{std}e^{\mu(T_{C}+T_{D})}}{2},V_{std}e^{\mu (T_{C}+T_{D})}]$ 10. The interval between division under the current environment. $\dfrac {ln2}{\mu}$ Section C: when will the normal cells be actually affected after blocking oriC When the normal bacteria are going to form a new replication fork and oriC is blocked at that time, the replication process is actually inhibited and normal cells become abnormal. $T_{r1}=\dfrac {ln2}{\mu}-xT_{C}. $ Equal with $T_{r1}=\dfrac {ln2}{\mu}-(\dfrac {lnV-lnV_{std}} {\mu}-\lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu})$ Section D: when will the abnormal cells recover to the normal stage. According to the state of bacteria under the effect of CRISPR/dCas9, cell volume will become much larger than to normal one, but it only have one chromosome. If we allow it initiates new replication process (release its OriC), it would initiate many OriC to recovery itself, but the previous model didn’t describe this phenomenon, so we define a new parameter “d” to make supplement. ASSUMPTION: Bacteria’s rule of growth: $V_{new} =V_{old}e^{\mu T}$

Bacteria’s rule of triggering replication innitition: $V>2^{N}V_{std}$

Bacteria will divide after phase C and phase Dcomplete, $T_{C}+T_{D}$ The trigger condition to initiate replication under abnormal situation: the shortest interval of forming two replication forks: d. if oriC is blocked within Tr1, the bacteria is not effected since the inhibition of Oric hasn’t actually change the state of bacteria. If oriC is blocked longer than Tr1, we have several analysis: The constraint condition of recovering to normal: $\dfrac{lnV-lnV_{std}}{\mu}-N\dfrac{ln2}{\mu}-k(\dfrac{ln2}{\mu}-d) <0$ K means the number of replication forks, d means minimum folk formation interval, N means the number of replication forks exist in the bacteria. The above formula is transformed, so that it can be solved directly: $k>\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}$ Because k is an integer, the formula add. so it’s better for computer calculation $k=\lceil\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}\rceil$ We get the mean of k, the dividing time for the bacteria to recover to normal is: N+k Finally calculate the time for bacteria to recover from abnormal to normal after oriC is released. $T_{r2}=T_{C}+T_{D}+d(k-1)$ It is worth notice that when the number of bacterial liabilities is less than 2, the minimum interval between two bifurcation must be less than ln2 / u, otherwise the bacteria will be dragged down by the new debt on the way of repayment and will not return to normal. Section E: evidence from experiment aTc has being added for 143min. At this time, the bacteria has passed through the Ton period in the model and aTc has already affected replication initiation. So an ultralong bacteria occurred. The temperature will be heated to 45℃ after 85min After being heated for 55min, the longest bacteria in the field of view is under the Tr2 stage. (At this time, there is no aTc in LB medium in the Microfluidic chip).this bacteria is recovering. The initiate stage is remained in the C period after dCas9 is released. After being heated for 133min, the ultralong bacteria has divided many times and recovered to the normal stage. Dividing many times in a short time also proved the validity of the model in a certain degree, which means the attribute “d” exists. We have realized this point before doing Microfluidic experiments through the model, and built mathematical model above. Approximate and preliminary measurement result of "d": 18 minutes. Future work: real-time fitting strategy Model integrates into software. Software controls hardware. Hardware regulates optogenetic system. Optogenetic system regulates growth rate. Growth rate affects the shape of bacteria. The shape of bacteria is analyzed by model. References 1. Stephen Cooper (2006). Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research. Theoretical Biology and Medical Modelling, 3:10 2. M Wallden, D Fange, EG Lundius, Ö Baltekin, J Elf (2016). The Synchronization of Replication and Division Cycles in Individual E.coli Cells. Cell, 166(3):729-739. 3. Cooper S, Helmstetter CE (1968). Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol 31(3):519–540.