Team:HZAU-China/Team
Model
For control cell cycle, our wet lab sign genetic circuit to control initiation location -- OriC. Finally Replication
initiation switch. Finally, we will get the bacterium with Replication initiation switch. It’s obviously better chosing
Oric as the controlled target cause the physiological states of germs won’t be severely effected compared with being
disposed with starvation method. But it also brings us two questions:
① How to observe OriC situation in time
After Dry Lab’s analysis, we can tell you that switch OriC is enough to control cell cycle, even better. Because we can
create more new status such as one E,coli has 8 volume but has 1 DNA.
② How to observe OriC situation in time?
Cause of the limiting of our lab, we couldn’t observe the OriC in single cell directly. So Dry Lab build a model based
on a series of previous published rules to describe the coordination among chromosome replication, cell growth and
division. And now we original catch three goals: ①improve it to Mathematical formula level. ② find its inaptitude during
repressed ,define a new parameter “d” to supply it. ③ only needs one kind data so easy to obtain--Volume,enough to
calculate E.coli’s every replication process (OriC situation) and other cell cycle information in single cell level.
In the process of modeling we encounter many challenges, such as Bacteria initiates multi-replication process during
one cycle.
Figure 1. The different stages of synchronization. Ton: the light triggering the synchronization system is on. And CRISPER
system will block Oric. Tri: 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 bacterium’s returning to normal. Toff: the time it takes from turning of the light
to Oric is unblocked. Tr2: time it takes for the bacteria returning to the normal state since Oric is unblocked.
Volume and replication model
There’s a lot of attributes for a single bacteria such as volume, weight, concentration of different kinds of proteins
and RNA, the amount of DNA and etc.but the accurate concentration of the matters can’t be directly detected due to
limited experimental conditions.Which means the variates and the concentration data the genetic circuit model needed
can’t de detected.This is a huge obstacle for the model’s accuracy.After thinking carefully, we summarized that through
the concentration of dCas9 and gRNA of the single bacteria can’t be directly acquired, the volume of which is the easiest
parameter we can get with Microfluidic photography and Flow cytometry that our project can be evaluated on single cellular
level other than bacterial 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 and made it formulized and softwarelized.
1.related basic rules
We checked some papers in the relevant research fields and summarized some rules of bacterium’s replication growth and
division, And applied them on our project.
bac’s rule of growth
$V_{new} =V_{old}e^{\mu T}$
bac’s rule of triggering replication innitition
$V>2^{N}V_{std}$
reason of division: bac will definitely divide after complete
$T_{C}+T_{D}$
Figure 3. diagrammatic sketch of the coordination among chromosome replication, cell growth and division
2. inferring the progress flow inner the bac
① Take the bacteria’s photo to get the volume Vold, Wait for the twait and take one photo again 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 out 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 the bac is V
④ The volume when the bac complete stage C and getting into stage D: $V_{D} =V_{std}e^{\mu T}$ ⑤ the progress of the smallest replication folk $$x= \begin{cases} \dfrac {lnV-lnV_{std}} {\mu T_{c}}- \lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu T_{C}} ,V>V_{std}\\ \\ 0,V < V_{std} \end{cases} $$ ⑥ Replication process of the replication folks $$\begin{cases} \quad \ \ \ x\\ \\ \dfrac {1} {log_{2}e^{\mu T_{c}}} + x\\ \\ \dfrac {2} {log_{2}e^{\mu T_{c}}} + x\\ \\ \quad \ \ \ .\\ \quad \ \ \ .\\ \quad \ \ \ .\\ \\ \dfrac {N_(max)} {log_{2}e^{\mu T_{c}}} + x \end{cases} $$ ⑦ Infer how long will the bac split $T_{C}+T_{D} - \dfrac {lnV-lnV_{std}}{\mu}$
⑧ Infer how long will it take for the bac to form the haploid chromosome without replication forks if all the Oric is blocked. $T_{C}+T_{D}-xT_{C}$
⑨ The range of volume cyclical changes: $[\dfrac {V_{std}e^{\mu(T_{C}+T_{D})}}{2},V_{std}e^{\mu (T_{C}+T_{D})}]$ ⑩ The interval between splits under the current grow environment. 3. when will the normal cells be really effected by Oric
Figure 4. When the normal bac is going to form a new replication fork and Oric is blocked at this time, the replication
process is really inhibited. so when Oric is unblocked, the time bac is actually effected is the time it takes
to form the new replication fork:
$T_{r1}=\dfrac {ln2}{\mu}-xT_{C}. $
Equal with
$T_{r1}=\dfrac {ln2}{\mu}-(\dfrac {lnV-lnV_{std}} {\mu}-\lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2}
{\mu})$
4. when will the abnormal cells recover to the normal, predictable stage.
According to the Bacteria under the CRISPR control, cell Volume will become many times to normal one, but it only
have one chromosome. If we allow its initiate new replication process (release its OriC), it want to initiate
many OriC to recovery itself, but previous model didn’t describe this phenom, so we define a new parameter “d”
to supply.
bac’s rule of growth
$V_{new} =V_{old}e^{\mu T}$
bac’s rule of triggering replication innitition $V>2^{N}V_{std}$
reason of division: bac will definitely divide after complete $T_{C}+T_{D}$ The triggercondition to initiate replication under abnormal situation: the shortest interval of forming two replication forks. ①The time needs to wait is 0 if Oric is unblocked in Tr1 since the inhibition of Oric hasn’t cause real effect to bac ②If Oric is unblocked after Tr1, we have those analysis: the restrict limits of becoming backe to normal: $\dfrac{lnV-lnV_{std}}{\mu}-N\dfrac{ln2}{\mu}-k(\dfrac{ln2}{\mu}-d) <0$ K means the number of replication folks, d means minimum folk forming interval, N means the replication folks exists in the bac. 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 bac to become normal is: N+k Finally the time needed for the bac to recover to normal from the blocking method is removed. $T_{r2}=T_{C}+T_{D}+d(k-1)$ It is worth noting 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. 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.
Figure 1. The different stages of synchronization. Ton: the light triggering the synchronization system is on. And CRISPER
system will block Oric. Tri: 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 bacterium’s returning to normal. Toff: the time it takes from turning of the light
to Oric is unblocked. Tr2: time it takes for the bacteria returning to the normal state since Oric is unblocked.
Volume and replication model
There’s a lot of attributes for a single bacteria such as volume, weight, concentration of different kinds of proteins
and RNA, the amount of DNA and etc.but the accurate concentration of the matters can’t be directly detected due to
limited experimental conditions.Which means the variates and the concentration data the genetic circuit model needed
can’t de detected.This is a huge obstacle for the model’s accuracy.After thinking carefully, we summarized that through
the concentration of dCas9 and gRNA of the single bacteria can’t be directly acquired, the volume of which is the easiest
parameter we can get with Microfluidic photography and Flow cytometry that our project can be evaluated on single cellular
level other than bacterial 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 and made it formulized and softwarelized.
1.related basic rules
We checked some papers in the relevant research fields and summarized some rules of bacterium’s replication growth and
division, And applied them on our project.
bac’s rule of growth
$V_{new} =V_{old}e^{\mu T}$
bac’s rule of triggering replication innitition
$V>2^{N}V_{std}$
reason of division: bac will definitely divide after complete
$T_{C}+T_{D}$
Figure 3. diagrammatic sketch of the coordination among chromosome replication, cell growth and division
2. inferring the progress flow inner the bac
① Take the bacteria’s photo to get the volume Vold, Wait for the twait and take one photo again 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 out 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 the bac is V
④ The volume when the bac complete stage C and getting into stage D: $V_{D} =V_{std}e^{\mu T}$ ⑤ the progress of the smallest replication folk $$x= \begin{cases} \dfrac {lnV-lnV_{std}} {\mu T_{c}}- \lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu T_{C}} ,V>V_{std}\\ \\ 0,V < V_{std} \end{cases} $$ ⑥ Replication process of the replication folks $$\begin{cases} \quad \ \ \ x\\ \\ \dfrac {1} {log_{2}e^{\mu T_{c}}} + x\\ \\ \dfrac {2} {log_{2}e^{\mu T_{c}}} + x\\ \\ \quad \ \ \ .\\ \quad \ \ \ .\\ \quad \ \ \ .\\ \\ \dfrac {N_(max)} {log_{2}e^{\mu T_{c}}} + x \end{cases} $$ ⑦ Infer how long will the bac split $T_{C}+T_{D} - \dfrac {lnV-lnV_{std}}{\mu}$
⑧ Infer how long will it take for the bac to form the haploid chromosome without replication forks if all the Oric is blocked. $T_{C}+T_{D}-xT_{C}$
⑨ The range of volume cyclical changes: $[\dfrac {V_{std}e^{\mu(T_{C}+T_{D})}}{2},V_{std}e^{\mu (T_{C}+T_{D})}]$ ⑩ The interval between splits under the current grow environment. 3. when will the normal cells be really effected by Oric
Figure 4. When the normal bac is going to form a new replication fork and Oric is blocked at this time, the replication
process is really inhibited. so when Oric is unblocked, the time bac is actually effected is the time it takes
to form the new replication fork:
$T_{r1}=\dfrac {ln2}{\mu}-xT_{C}. $
Equal with
$T_{r1}=\dfrac {ln2}{\mu}-(\dfrac {lnV-lnV_{std}} {\mu}-\lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2}
{\mu})$
4. when will the abnormal cells recover to the normal, predictable stage.
According to the Bacteria under the CRISPR control, cell Volume will become many times to normal one, but it only
have one chromosome. If we allow its initiate new replication process (release its OriC), it want to initiate
many OriC to recovery itself, but previous model didn’t describe this phenom, so we define a new parameter “d”
to supply.
bac’s rule of growth
$V_{new} =V_{old}e^{\mu T}$
bac’s rule of triggering replication innitition $V>2^{N}V_{std}$
reason of division: bac will definitely divide after complete $T_{C}+T_{D}$ The triggercondition to initiate replication under abnormal situation: the shortest interval of forming two replication forks. ①The time needs to wait is 0 if Oric is unblocked in Tr1 since the inhibition of Oric hasn’t cause real effect to bac ②If Oric is unblocked after Tr1, we have those analysis: the restrict limits of becoming backe to normal: $\dfrac{lnV-lnV_{std}}{\mu}-N\dfrac{ln2}{\mu}-k(\dfrac{ln2}{\mu}-d) <0$ K means the number of replication folks, d means minimum folk forming interval, N means the replication folks exists in the bac. 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 bac to become normal is: N+k Finally the time needed for the bac to recover to normal from the blocking method is removed. $T_{r2}=T_{C}+T_{D}+d(k-1)$ It is worth noting 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. 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.
