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First Authorlabel=e1]first@somewhere.com [    Second Authorlabel=e2]second@somewhere.com [ Address of the First and Second authors
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(2020)
Abstract

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and

t1Some comment t2First supporter of the project t3Second supporter of the project

1 Ordinary text

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2 Notes

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All conjectures are interesting, but some conjectures are more interesting than others.

Proof.

Obvious. ∎

4 Tables and figures

Cross reference to labelled table: As you can see in Table 1 on page 1 and also in Table 2 on page 2.

Table 1: The spherical case (I1=0I_{1}=0, I2=0I_{2}=0).
Equil.
Points xx yy zz CC S
L1~{}~{}L_{1} -2.485252241 0.000000000 0.017100631 8.230711648 U
L2~{}~{}L_{2} 0.000000000 0.000000000 3.068883732 0.000000000 S
L3~{}~{}L_{3} 0.009869059 0.000000000 4.756386544 -0.000057922 U
L4~{}~{}L_{4} 0.210589855 0.000000000 -0.007021459 9.440510897 U
L5~{}~{}L_{5} 0.455926604 0.000000000 -0.212446624 7.586126667 U
L6~{}~{}L_{6} 0.667031314 0.000000000 0.529879957 3.497660052 U
L7~{}~{}L_{7} 2.164386674 0.000000000 -0.169308438 6.866562449 U
L8~{}~{}L_{8} 0.560414471 0.421735658 -0.093667445 9.241525367 U
L9~{}~{}L_{9} 0.560414471 -0.421735658 -0.093667445 9.241525367 U
L10~{}~{}L_{10} 1.472523232 1.393484549 -0.083801333 6.733436505 U
L11~{}~{}L_{11} 1.472523232 -1.393484549 -0.083801333 6.733436505 U

A major point of difference lies in the value of the specific production rate π\pi for large values of the specific growth rate μ\mu. Already in the early publications [1, 2, 3] it appeared that high glucose concentrations in the production phase are well correlated with a low penicillin yield (the ‘glucose effect’). It has been confirmed recently [1, 2, 3, 4] that high glucose concentrations inhibit the synthesis of the enzymes of the penicillin pathway, but not the actual penicillin biosynthesis. In other words, glucose represses (and not inhibits) the penicillin biosynthesis.

These findings do not contradict the results of [1] and of [4] which were obtained for continuous culture fermentations. Because for high values of the specific growth rate μ\mu it is most likely (as shall be discussed below) that maintenance metabolism occurs, it can be shown that in steady state continuous culture conditions, and with μ\mu described by a Monod kinetics

Cs=KMμ/μx1μ/μxC_{s}=K_{M}\frac{\mu/\mu_{x}}{1-\mu/\mu_{x}} (4.1)

Pirt & Rhigelato determined π\pi for μ\mu between 0.0230.023 and 0.0860.086 h-1. They also reported a value μx0.095\mu_{x}\approx 0.095 h-1, so that for their experiments μ/μx\mu/\mu_{x} is in the range of 0.240.24 to 0.90.9. Substituting KMK_{M} in Eq. (4.1) by the value KM=1K_{M}=1 g/L as used by [1], one finds with the above equation 0.3<Cs<90.3<C_{s}<9 g/L. This agrees well with the work of [4], who reported that penicillin biosynthesis repression only occurs at glucose concentrations from Cs=10C_{s}=10 g/L on. The conclusion is that the glucose concentrations in the experiments of Pirt & Rhigelato probably were too low for glucose repression to be detected. The experimental data published by Ryu & Hospodka are not detailed sufficiently to permit a similar analysis.

Table 2: Parameter sets used by Bajpai & Reuß 
parameter Set 1 Set 2
μx\mu_{x} [h-1] 0.092 0.11
KxK_{x} [g/g DM] 0.15 0.006
μp\mu_{p} [g/g DM h] 0.005 0.004
KpK_{p} [g/L] 0.0002 0.0001
KiK_{i} [g/L] 0.1 0.1
Yx/sY_{x/s} [g DM/g] 0.45 0.47
Yp/sY_{p/s} [g/g] 0.9 1.2
khk_{h} [h-1] 0.04 0.01
msm_{s} [g/g DM h] 0.014 0.029

Bajpai & Reuß decided to disregard the differences between time constants for the two regulation mechanisms (glucose repression or inhibition) because of the relatively very long fermentation times, and therefore proposed a Haldane expression for π\pi.

It is interesting that simulations with the [4] model for the initial conditions given by these authors indicate that, when the remaining substrate is fed at a constant rate, a considerable and unrealistic amount of penicillin is produced when the glucose concentration is still very high [2, 3, 4] Simulations with the Bajpai & Reuß model correctly predict almost no penicillin production in similar conditions.

Refer to caption
Figure 1: Example of figure inclusion.

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5 Headings

5.1 Subsection

Carr-Goldstein based their model on balancing methods and biochemical knowledge. The original model (1980) contained an equation for the oxygen dynamics which has been omitted in a second paper (1981). This simplified model shall be discussed here.

5.1.1 Subsubsection

Carr-Goldstein based their model on balancing methods and biochemical knowledge. The original model (1980) contained an equation for the oxygen dynamics which has been omitted in a second paper (1981). This simplified model shall be discussed here.

6 Equations and the like

Two equations:

Cs=KMμ/μx1μ/μxC_{s}=K_{M}\frac{\mu/\mu_{x}}{1-\mu/\mu_{x}} (6.1)

and

G=PoptPrefPref 100 (%)G=\frac{P_{\rm opt}-P_{\rm ref}}{P_{\rm ref}}\mbox{\ }100\mbox{\ }(\%) (6.2)

Two equation arrays:

dSdt\displaystyle\frac{dS}{dt} =\displaystyle= σX+sFF\displaystyle-\sigma X+s_{F}F (6.3)
dXdt\displaystyle\frac{dX}{dt} =\displaystyle= μX\displaystyle\mu X (6.4)
dPdt\displaystyle\frac{dP}{dt} =\displaystyle= πXkhP\displaystyle\pi X-k_{h}P (6.5)
dVdt\displaystyle\frac{dV}{dt} =\displaystyle= F\displaystyle F (6.6)

and,

μsubstr\displaystyle\mu_{\rm substr} =\displaystyle= μxCsKxCx+Cs\displaystyle\mu_{x}\frac{C_{s}}{K_{x}C_{x}+C_{s}} (6.7)
μ\displaystyle\mu =\displaystyle= μsubstrYx/s(1H(Cs))(ms+π/Yp/s)\displaystyle\mu_{\rm substr}-Y_{x/s}(1-H(C_{s}))(m_{s}+\pi/Y_{p/s}) (6.8)
σ\displaystyle\sigma =\displaystyle= μsubstr/Yx/s+H(Cs)(ms+π/Yp/s)\displaystyle\mu_{\rm substr}/Y_{x/s}+H(C_{s})(m_{s}+\pi/Y_{p/s}) (6.9)

Appendix A Appendix section

We consider a sequence of queueing systems indexed by nn. It is assumed that each system is composed of JJ stations, indexed by 11 through JJ, and KK customer classes, indexed by 11 through KK. Each customer class has a fixed route through the network of stations. Customers in class kk, k=1,,Kk=1,\ldots,K, arrive to the system according to a renewal process, independently of the arrivals of the other customer classes. These customers move through the network, never visiting a station more than once, until they eventually exit the system.

A.1 Appendix subsection

However, different customer classes may visit stations in different orders; the system is not necessarily “feed-forward.” We define the path of class kk customers in as the sequence of servers they encounter along their way through the network and denote it by

𝒫=(jk,1,jk,2,,jk,m(k)).\mathcal{P}=\bigl{(}j_{k,1},j_{k,2},\dots,j_{k,m(k)}\bigr{)}. (A.1)

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{acks}

[Acknowledgments] And this is an acknowledgements section with a heading that was produced by the \\backslashsection* command. Thank you all for helping me writing this  sample file.

{supplement}\stitle

Title of Supplement A \sdescriptionShort description of Supplement A. {supplement} \stitleTitle of Supplement B \sdescriptionShort description of Supplement B.

References

  • [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York. \MR1700749
  • [2] Bourbaki, N. (1966). General Topology 1. Addison–Wesley, Reading, MA.
  • [3] Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. Wiley, New York. \MR838085
  • [4] Prokhorov, Yu. (1956). Convergence of random processes and limit theorems in probability theory. Theory Probab. Appl. 1 157–214. \MR84896