Difference between revisions of "Team:HZAU-China/Model"

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<body>
 
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   <div class="HZAU_div_main">
 
   <div class="HZAU_div_main">
     <a class="biaoti">Model</a>
+
     <a class="HZAU_title">Model</a>
     <a class="zhengwen">For control cell cycle, our wet lab sign genetic circuit to control initiation location -- OriC. Finally Replication
+
     <a class="zhengwen">To control cell cycle, we designed a genetic circuit to control replication by blocking the replication initiation site, oriC. But it brings us two questions:</a>
      initiation switch. Finally, we will get the bacterium with Replication initiation switch. It’s obviously better chosing
+
     <a class="zhengwen">Is this approach effective enough to control cell cycle?</a>
      Oric as the controlled target cause the physiological states of germs won’t be severely effected compared with being
+
     <a class="zhengwen">After Dry Lab’s analysis, we found that blocking oriC is well enough to control cell cycle, and even better since. Because we can create new conditions of E.coli that has not been discovered such as E.,coli with 8-fold volume but only has one genome. </a>
      disposed with starvation method. But it also brings us two questions:</a>
+
     <a class="zhengwen">How to observe replication procession in real-time?</a>
     <a class="zhengwen">① How to observe OriC situation in time</a>
+
     <a class="zhengwen">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 come 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.  
     <a class="zhengwen">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.</a>
+
     <a class="zhengwen">How to observe OriC situation in time?</a>
+
     <a class="zhengwen">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.
+
 
     </a>
 
     </a>
     <a class="zhengwen">In the process of modeling we encounter many challenges, such as Bacteria initiates multi-replication process during
+
     <a class="zhengwen">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 mathematical formulation, but with some smart trick we can formulate it into a mathematical function. </a>
      one cycle.</a>
+
     <img src="https://static.igem.org/mediawiki/2017/2/2e/Model1.png" class="tu_1">
     <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
+
     <a class="zhengwen">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. </a>
     <a class="zhengwen">Figure 1. The different stages of synchronization. Ton: the light triggering the synchronization system is on. And CRISPER
+
     <img src="https://static.igem.org/mediawiki/2017/7/73/T--HZAU-China--model2.png" class="tu_1">
      system will block Oric. Tri: It won’t exert any effect when Oric is not blocked since it can’t initiate new replication
+
     <a class="zhengwen">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.</a>
      fork. Ts:time of becoming single chromosome.</a>
+
     <a class="zhengwen">Section A Volume and replication model</a>
     <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
+
     <a class="zhengwen">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 the matters can’t be directly detected due to limited experimental condition. Which means the variates and the concentration of the genetic circuit can’t detected. This is a huge obstacle for the modeling’s accuracy. After thinking carefully, we summarized that though 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. So 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.
     <a class="zhengwen">Figure 2. The different stages of the bacterium’s returning to normal. Toff: the time it takes from turning of the light
+
          </a>
      to Oric is unblocked. Tr2: time it takes for the bacteria returning to the normal state since Oric is unblocked.</a>
+
     <a class="zhengwen"> 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:</a>
     <a class="zhengwen">Volume and replication model</a>
+
   
     <a class="zhengwen">There’s a lot of attributes for a single bacteria such as volume, weight, concentration of different kinds of proteins
+
     <a class="zhengwen_disblock">Bacterial rule of growth: </a>
      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.</a>
+
     <a class="zhengwen">Thus we improved our previous work and made it formulized and softwarelized.</a>
+
    <a class="zhengwen">1.related basic rules</a>
+
    <a class="zhengwen">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.</a>
+
     <a class="zhengwen_disblock">bac’s rule of growth</a>
+
 
     <a>$V_{new} =V_{old}e^{\mu T}$</a>
 
     <a>$V_{new} =V_{old}e^{\mu T}$</a>
     <a class="zhengwen_disblock">bac’s rule of triggering replication innitition</a>
+
     <a class="zhengwen_disblock">Bacterial rule of triggering replication initiation: </a>
 
     <a>$V>2^{N}V_{std}$</a>
 
     <a>$V>2^{N}V_{std}$</a>
     <a class="zhengwen_disblock">reason of division: bac will definitely divide after complete</a>
+
     <a class="zhengwen_disblock">Bacteria will divide after C phase and D phase: </a>
 
     <a class="HZAU_gongshi">$T_{C}+T_{D}$</a>
 
     <a class="HZAU_gongshi">$T_{C}+T_{D}$</a>
 
     <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
 
     <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
 
     <a class="zhengwen">Figure 3. diagrammatic sketch of the coordination among chromosome replication, cell growth and division</a>
 
     <a class="zhengwen">Figure 3. diagrammatic sketch of the coordination among chromosome replication, cell growth and division</a>
     <a class="zhengwen">2. inferring the progress flow inner the bac</a>
+
     <a class="zhengwen">Section B inferring the progress of replication inner the bacteria</a>
     <a class="zhengwen">Take the bacteria’s photo to get the volume Vold, Wait for the twait and take one photo again to get the volume Vnew
+
     <a class="zhengwen">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
 
     </a>
 
     </a>
     <a class="zhengwen_disblock">Calculate the growth rate according to the formula.</a>
+
     <a class="zhengwen_disblock">2. Calculate the growth rate according to the formula.</a>
 
     <a class="HZAU_gongshi">$\mu =\dfrac {nV_{nen}-lnV_{old}} {t_{wait}}$</a>
 
     <a class="HZAU_gongshi">$\mu =\dfrac {nV_{nen}-lnV_{old}} {t_{wait}}$</a>
 
     <a class="zhengwen_disblock">
 
     <a class="zhengwen_disblock">
       <br>③ it can be inferred out that the upper limit of the replication folk Nmax is </a>
+
       <br>3. It can be inferred that the upper limit of the replication folk Nmax is </a>
 
     <a class="HZAU_gongshi">$\lceil log_{2}V_{std}e^{\mu(T_{C}+T_{D})} \rceil$
 
     <a class="HZAU_gongshi">$\lceil log_{2}V_{std}e^{\mu(T_{C}+T_{D})} \rceil$
       <a class="zhengwen_disblock">when the volume of the bac is V</a>
+
       <a class="zhengwen_disblock">when the volume of bacteria is V.</a>
 
       <a class="zhengwen_disblock">
 
       <a class="zhengwen_disblock">
         <br>The volume when the bac complete stage C and getting into stage D:</a>
+
         <br>4. The volume when the bacteria complete stage C and getting into stage D: </a>
 
       <a class="HZAU_gongshi">$V_{D} =V_{std}e^{\mu T}$</a>
 
       <a class="HZAU_gongshi">$V_{D} =V_{std}e^{\mu T}$</a>
       <a class="zhengwen">the progress of the smallest replication folk</a>
+
       <a class="zhengwen">5. the progress of the smallest replication folk</a>
 
       <a class="HZAU_gongshi">$$x= \begin{cases} \dfrac {lnV-lnV_{std}} {\mu T_{c}}- \lfloor \dfrac {lnV-lnV_{std}} {ln2} \rfloor \dfrac {ln2} {\mu
 
       <a class="HZAU_gongshi">$$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
 
         T_{C}} ,V>V_{std}\\ \\ 0,V
 
         < V_{std} \end{cases} $$</a>
 
         < V_{std} \end{cases} $$</a>
           <a class="zhengwen">Replication process of the replication folks</a>
+
        <a class="zhengwen">Note: x is greater than 100% means the bac complete stage C and reached stage D.</a>
 +
           <a class="zhengwen">6. Replication process of the replication forks</a>
 
           <a class="HZAU_gongshi">$$\begin{cases} \quad \ \ \ x\\ \\ \dfrac {1} {log_{2}e^{\mu T_{c}}} + x\\ \\ \dfrac {2} {log_{2}e^{\mu T_{c}}}
 
           <a class="HZAU_gongshi">$$\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}
 
             + x\\ \\ \quad \ \ \ .\\ \quad \ \ \ .\\ \quad \ \ \ .\\ \\ \dfrac {N_(max)} {log_{2}e^{\mu T_{c}}} + x \end{cases}
 
             $$
 
             $$
 
           </a>
 
           </a>
           <a class="zhengwen_disblock">⑦ Infer how long will the bac split</a>
+
           <a class="zhengwen_disblock">7. Inferring how long will the bacteria divide:</a>
 
           <a class="HZAU_gongshi">$T_{C}+T_{D} - \dfrac {lnV-lnV_{std}}{\mu}$</a>
 
           <a class="HZAU_gongshi">$T_{C}+T_{D} - \dfrac {lnV-lnV_{std}}{\mu}$</a>
 
           <a class="zhengwen_disblock">
 
           <a class="zhengwen_disblock">
             <br>⑧ Infer how long will it take for the bac to form the haploid chromosome without replication forks if all the
+
             <br>8. Inferring how long will it take for the bacteria to form the haploid chromosome if all the Oric is blocked.</a>
            Oric is blocked.</a>
+
 
           <a class="HZAU_gongshi">$T_{C}+T_{D}-xT_{C}$</a>
 
           <a class="HZAU_gongshi">$T_{C}+T_{D}-xT_{C}$</a>
 
           <a class="zhengwen_disblock">
 
           <a class="zhengwen_disblock">
             <br>The range of volume cyclical changes:</a>
+
             <br>9. The range of periodically changed volume</a>
 
           <a class="HZAU_gongshi">$[\dfrac {V_{std}e^{\mu(T_{C}+T_{D})}}{2},V_{std}e^{\mu (T_{C}+T_{D})}]$</a>
 
           <a class="HZAU_gongshi">$[\dfrac {V_{std}e^{\mu(T_{C}+T_{D})}}{2},V_{std}e^{\mu (T_{C}+T_{D})}]$</a>
           <a class="zhengwen">The interval between splits under the current grow environment.</a>
+
           <a class="zhengwen">10. The interval between division under the current environment.</a>
           <a class="zhengwen">3. when will the normal cells be really effected by Oric</a>
+
          <a class="HZAU_gongshi">$\dfrac {ln2}{\mu}$</a>
 +
           <a class="zhengwen">Section C when will the normal cells be actually affected after blocking oriC</a>
 
           <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
 
           <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
           <a class="zhengwen_disblock">Figure 4. When the normal bac is going to form a new replication fork and Oric is blocked at this time, the replication
+
           <a class="zhengwen_disblock">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.</a>
            process is really inhibited. so when Oric is unblocked, the time bac is actually effected is the time it takes
+
          <a class="zhengwen_disblock"So when oriC is blocked, the time when bacteria is actually affected is the time it takes to form a new replication fork.</a>
            to form the new replication fork:</a>
+
         
 
           <a class="HZAU_gongshi">$T_{r1}=\dfrac {ln2}{\mu}-xT_{C}. $</a>
 
           <a class="HZAU_gongshi">$T_{r1}=\dfrac {ln2}{\mu}-xT_{C}. $</a>
 
           <a class="zhengwen_disblock">Equal with</a>
 
           <a class="zhengwen_disblock">Equal with</a>
Line 261: Line 306:
 
             {\mu})$
 
             {\mu})$
 
           </a>
 
           </a>
           <a class="zhengwen">4. when will the abnormal cells recover to the normal, predictable stage.</a>
+
           <a class="zhengwen">Section D when will the abnormal cells recover to the normal stage.</a>
           <a class="zhengwen">According to the Bacteria under the CRISPR control, cell Volume will become many times to normal one, but it only
+
           <a class="zhengwen">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.</a>
            have one chromosome. If we allow its initiate new replication process (release its OriC), it want to initiate
+
          <a class="zhengwen">ASSUMPTION: </a>
            many OriC to recovery itself, but previous model didn’t describe this phenom, so we define a new parameter “d”
+
           <a class="zhengwen_disblock">Bacteria’s rule of growth: </a>
            to supply.</a>
+
           <a class="zhengwen_disblock">bac’s rule of growth</a>
+
 
           <a class="HZAU_gongshi">$V_{new} =V_{old}e^{\mu T}$</a>
 
           <a class="HZAU_gongshi">$V_{new} =V_{old}e^{\mu T}$</a>
           <a class="zhengwen_disblock">bac’s rule of triggering replication innitition</a>
+
           <a class="zhengwen_disblock"><br>Bacteria’s rule of triggering replication innitition: </a>
 
           <a class="HZAU_gongshi">$V>2^{N}V_{std}$</a>
 
           <a class="HZAU_gongshi">$V>2^{N}V_{std}$</a>
           <a class="zhengwen_disblock">reason of division: bac will definitely divide after complete</a>
+
           <a class="zhengwen_disblock"><br>Bacteria will divide after phase C and phase Dcomplete,</a>
 
           <a class="HZAU_gongshi">$T_{C}+T_{D}$</a>
 
           <a class="HZAU_gongshi">$T_{C}+T_{D}$</a>
           <a class="zhengwen">The triggercondition to initiate replication under abnormal situation: the shortest interval of forming two replication
+
           <a class="zhengwen">The trigger condition to initiate replication under abnormal situation: the shortest interval of forming two replication forks: d.
            forks.
+
 
           </a>
 
           </a>
           <a class="zhengwen">①The time needs to wait is 0 if Oric is unblocked in Tr1 since the inhibition of Oric hasn’t cause real effect
+
           <a class="zhengwen">if oriC is blocked within Tr1, the bacteria is not effected since the inhibition of Oric hasn’t acutually change the state of bacteria.</a>
            to bac</a>
+
           <a class="zhengwen">If oriC is blocked longer than Tr1, we have several analysis:
           <a class="zhengwen">②If Oric is unblocked after Tr1, we have those analysis:</a>
+
                The constraint condition of recovering to normal: </a>
          <a class="zhengwen">the restrict limits of becoming backe to normal:</a>
+
 
           <a class="HZAU_gongshi">$\dfrac{lnV-lnV_{std}}{\mu}-N\dfrac{ln2}{\mu}-k(\dfrac{ln2}{\mu}-d)
 
           <a class="HZAU_gongshi">$\dfrac{lnV-lnV_{std}}{\mu}-N\dfrac{ln2}{\mu}-k(\dfrac{ln2}{\mu}-d)
 
             <0$</a>
 
             <0$</a>
               <a class="zhengwen">K means the number of replication folks, d means minimum folk forming interval, N means the replication folks
+
               <a class="zhengwen">K means the number of replication forks, d means minimum folk formation interval, N means the number of replication forks exist in the bacteria.</a>
                exists in the bac.</a>
+
 
               <a class="zhengwen">The above formula is transformed, so that it can be solved directly:</a>
 
               <a class="zhengwen">The above formula is transformed, so that it can be solved directly:</a>
 
               <a class="HZAU_gongshi">$k>\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}$</a>
 
               <a class="HZAU_gongshi">$k>\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}$</a>
               <a class="zhengwen">Because k is an integer, the formula add *** so it’s better for computer calculation.</a>
+
               <a class="zhengwen">Because k is an integer, the formula add. so it’s better for computer calculation</a>
 
               <a class="HZAU_gongshi">$k=\lceil\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}\rceil$</a>
 
               <a class="HZAU_gongshi">$k=\lceil\dfrac{lnV-lnV_{std}-Nln2}{ln2-\mu d}\rceil$</a>
               <a class="zhengwen">We get the mean of k, the dividing time for the bac to become normal is:</a>
+
               <a class="zhengwen">We get the mean of k, the dividing time for the bacteria to recover to normal is: </a>
 
               <a class="HZAU_gongshi">N+k</a>
 
               <a class="HZAU_gongshi">N+k</a>
               <a class="zhengwen">Finally the time needed for the bac to recover to normal from the blocking method is removed.</a>
+
               <a class="zhengwen">Finally calculate the time for bacteria to recover from abnormal to normal after oriC is released.</a>
 
               <a class="HZAU_gongshi">$T_{r2}=T_{C}+T_{D}+d(k-1)$</a>
 
               <a class="HZAU_gongshi">$T_{r2}=T_{C}+T_{D}+d(k-1)$</a>
               <a class="zhengwen">It is worth noting that when the number of bacterial liabilities is less than 2, the minimum interval between
+
               <a class="zhengwen">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.</a>
                two bifurcation must be less than ln2 / u, otherwise the bacteria will be dragged down by the new debt on
+
              <a class="zhengwen">Section E evidence getting from experiment</a>
                the way of repayment and will not return to normal.</a>
+
              <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
 +
              <a class="zhengwen">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</a>
 +
              <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
 +
              <a class="zhengwen">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.</a>
 +
              <img src="https://static.igem.org/mediawiki/2017/4/44/T--HZAU-China--BCDperiod.png" class="tu_1">
 +
              <a class="zhengwen">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.</a>
 +
              <a class="fubiaoti">Future work: Realtime fitting strategy</a>
 +
 
 +
 
 +
              <a class="zhengwen">Model constructs into software. Software controls hardware. Hardware regulates optogenetic system. Optogenetic system regulates growth rate. Growth rate effects the shape of bacteria.</a>
 +
              <a class="zhengwen">The shape of bacteria is analyzed by model.</a>
 +
 
 +
 
 +
 
 +
 
 
                 <a class="biaoti">References</a>
 
                 <a class="biaoti">References</a>
                 <a class="yinwen">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. </a>
+
                 <a class="yinwen">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</a>
 
                 <a class="yinwen">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.</a>
 
                 <a class="yinwen">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.</a>
 
                 <a class="yinwen">3. Cooper S, Helmstetter CE (1968). Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol 31(3):519–540.</a>
 
                 <a class="yinwen">3. Cooper S, Helmstetter CE (1968). Chromosome replication and the division cycle of Escherichia coli B/r. J Mol Biol 31(3):519–540.</a>

Revision as of 03:41, 2 November 2017

Model To control cell cycle, we designed a genetic circuit to control replication by blocking the replication initiation site, oriC. But it brings us two questions: 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. Because we can create new conditions of E.coli that has not been discovered such as E.,coli with 8-fold volume but only has one genome. 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 come 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 mathematical formulation, 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. 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 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 the matters can’t be directly detected due to limited experimental condition. Which means the variates and the concentration of the genetic circuit can’t detected. This is a huge obstacle for the modeling’s accuracy. After thinking carefully, we summarized that though 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. So 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, 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 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 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 acutually 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 getting 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. Future work: Realtime fitting strategy Model constructs into software. Software controls hardware. Hardware regulates optogenetic system. Optogenetic system regulates growth rate. Growth rate effects 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.