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</settings> <regions> <region region-id="237" left="6" top="2.25" width="356.25" height="24" align-x="6" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Problem 9.2 of Shuler & Kargi.<sp count="2"/>Batch fermentor operated in two stages. Instructor: Nam Sun Wang </text> </region> <region region-id="238" left="6" top="30" width="436.5" height="9.75" align-x="6" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="2" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="3"/> </region> <region region-id="240" left="6" top="44.25" width="102.75" height="12" align-x="6" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Cell growth parameters: </text> </region> <region region-id="309" left="18" top="63" width="42" height="16.5" align-x="41.25" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="m1">μ</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="310" left="66" top="59.25" width="15" height="15" align-x="66" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> h-1 <sp/> </text> </region> <region region-id="311" left="108" top="63" width="35.25" height="16.5" align-x="131.25" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="m2">μ</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="312" left="174" top="63" width="36" height="16.5" align-x="191.25" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="s">K</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="313" left="222" top="62.25" width="24" height="12" align-x="222" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> g/liter </text> </region> <region region-id="288" left="270" top="63" width="34.5" height="16.5" align-x="285.75" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">Y</ml:id> 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item-idref="9"/> </region> <region region-id="283" left="6" top="116.25" width="131.25" height="12" align-x="6" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Product formation parameters: </text> </region> <region region-id="289" left="18" top="129" width="36" height="16.5" align-x="35.25" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">Y</ml:id> <ml:real>0.6</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="290" left="66" top="128.25" width="92.25" height="12" align-x="66" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> g product/g substrate </text> </region> <region region-id="291" left="168" top="129" width="39.75" height="16.5" align-x="184.5" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">q</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="303" left="216" top="128.25" width="80.25" height="12" align-x="216" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> g product/(g cell<f family="Symbol" charset="2"></f>h) </text> </region> <region region-id="298" left="6" top="146.25" width="64.5" height="12" align-x="6" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Initial condition </text> </region> <region region-id="302" left="84" top="147" width="33" height="16.5" align-x="98.25" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">x</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="301" left="132" top="147" width="27" height="16.5" align-x="147" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id 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background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Dynamic equations for the second phase of batch fermentor. </text> </region> <region region-id="80" left="6" top="327" width="145.5" height="16.5" align-x="54" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">dx2dt</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> 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<math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">dp2dt</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve" subscript="p">q</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="83" left="6" top="393" width="159" height="61.5" align-x="72" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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xml:space="preserve">xsp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="327" left="6" top="458.25" width="129.75" height="12" align-x="6" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Integrate 2 ODEs sequentially </text> </region> <region region-id="69" left="6" top="477" width="265.5" height="98.25" align-x="61.5" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">txsp</ml:id> <ml:boundVars> <ml:id xml:space="preserve" 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<ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="381" left="0" top="576" width="6000" height="6" align-x="0" align-y="576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="362" left="6" top="578.25" width="129" height="12" align-x="6" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Maximize product productivity </text> </region> <region region-id="343" left="6" top="597" width="60.75" height="16.5" align-x="22.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.1</ml:real> </ml:sequence> <ml:real>15</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="345" left="6" top="612" width="206.25" height="159" align-x="6" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="31"/> <rendering item-idref="32"/> </region> <region region-id="366" left="6" top="770.25" width="86.25" height="12" align-x="6" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Provide initial guess </text> </region> <region region-id="367" left="108" top="771" width="33" height="16.5" align-x="124.5" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="368" left="162" top="771" width="33" height="16.5" align-x="178.5" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f2">t</ml:id> <ml:real>50</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="369" left="210" top="765" width="122.25" height="22.5" align-x="267" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval 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xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.022929483384865081</ml:real> </result> </ml:eval> </math> <rendering item-idref="37"/> </region> <region region-id="380" left="192" top="818.25" width="244.5" height="12" align-x="192" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The productivity in 2 CSTRs in series was 0.027 g/(L<f family="Symbol" charset="2"></f>h) </text> </region> <region region-id="373" left="6" top="836.25" width="25.5" height="12" align-x="6" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> profile </text> </region> <region region-id="374" left="42" top="837" width="79.5" height="16.5" align-x="64.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">txsp</ml:id> <ml:apply> <ml:id xml:space="preserve">txsp</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="376" left="144" top="837.75" width="201" height="18" align-x="196.5" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="1" cols="4"> <ml:id 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height="9.75" align-x="6" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="42" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="43"/> </region> <region region-id="393" left="6" top="1010.25" width="432" height="24" align-x="6" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The product is encoded in a plasmid. 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width="71.25" height="16.5" align-x="65.25" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="dt">dp1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="511" left="0" top="1140" width="6000" height="6" align-x="0" align-y="1140" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="395" left="6" top="1143.75" width="207.75" height="83.25" align-x="82.5" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="dt">dxysp1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">xysp</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="4" cols="1"> <ml:apply> <ml:id xml:space="preserve" subscript="dt">dx1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="dt">dy1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id 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tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="dt">dxysp2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">xysp</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="4" cols="1"> <ml:apply> <ml:id xml:space="preserve" subscript="dt">dx2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="dt">dy2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="dt">ds2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="dt">dp2</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xysp</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:matrix> </ml:define> </math> 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Fermentation time for 5-h induction was longer than that for 10-h induction (~32h versus ~20h). Product concentration for 5-h induction was lower than that for 10-h induction (0.235 g/L versus 0.355 g/L).<sp count="2"/>5-h induction led to a fermentor full of nonproductive plasmid-free cells, the fraction of nonproductive plasmid-free cells being 0.777 for 5-h induction versus 0.022 for 10-h induction. </text> </region> <region region-id="556" left="6" top="2198.25" width="80.25" height="12" align-x="6" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Sub optimal profile </text> </region> <region region-id="557" left="102" top="2199" width="33" height="16.5" align-x="118.5" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" 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random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="error"> <compatibility multiple-assignment="MC12" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="632" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="237" left="6" top="2.25" width="356.25" height="24" align-x="6" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Problem 9.2 of Shuler & Kargi.<sp count="2"/>Batch fermentor operated in two stages. Instructor: Nam Sun Wang </text> </region> <region region-id="238" left="6" top="30" width="436.5" height="9.75" align-x="6" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="2" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="3"/> </region> <region region-id="240" left="6" top="44.25" width="102.75" height="12" align-x="6" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Cell growth parameters: </text> </region> <region region-id="309" left="18" top="63" width="42" height="16.5" align-x="41.25" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="m1">μ</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="310" left="66" top="59.25" width="15" height="15" align-x="66" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> h-1 <sp/> </text> </region> <region region-id="311" left="108" top="63" width="35.25" height="16.5" align-x="131.25" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="m2">μ</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="312" left="174" top="63" width="36" height="16.5" align-x="191.25" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="s">K</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="313" left="222" top="62.25" width="24" height="12" align-x="222" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> g/liter </text> </region> <region region-id="288" left="270" top="63" width="34.5" height="16.5" align-x="285.75" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="x">Y</ml:id> 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item-idref="9"/> </region> <region region-id="283" left="6" top="116.25" width="131.25" height="12" align-x="6" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Product formation parameters: </text> </region> <region region-id="289" left="18" top="129" width="36" height="16.5" align-x="35.25" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">Y</ml:id> <ml:real>0.6</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="290" left="66" top="128.25" width="92.25" height="12" align-x="66" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> g product/g substrate </text> </region> <region region-id="291" left="168" top="129" width="39.75" height="16.5" align-x="184.5" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">q</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="303" left="216" top="128.25" width="80.25" height="12" align-x="216" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> g product/(g cell<f family="Symbol" charset="2"></f>h) </text> </region> <region region-id="298" left="6" top="146.25" width="64.5" height="12" align-x="6" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Initial condition </text> </region> <region region-id="302" left="84" top="147" width="33" height="16.5" align-x="98.25" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="0">x</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="301" left="132" top="147" width="27" height="16.5" align-x="147" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id 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<math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">dp2dt</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:id xml:space="preserve" subscript="p">q</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="83" left="6" top="393" width="159" height="61.5" align-x="72" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">dxsp2dt</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">xsp</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:id xml:space="preserve">dx2dt</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">ds2dt</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">dp2dt</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">xsp</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="327" left="6" top="458.25" width="129.75" height="12" align-x="6" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Integrate 2 ODEs sequentially </text> </region> <region region-id="69" left="6" top="477" width="265.5" height="98.25" align-x="61.5" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">txsp</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:boundVars> </ml:function> <ml:program> <ml:localDefine> <ml:id xml:space="preserve">txsp1</ml:id> <ml:apply> <ml:id xml:space="preserve">rkfixed</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="0">xsp</ml:id> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">dxs1dt</ml:id> </ml:sequence> </ml:apply> </ml:localDefine> <ml:localDefine> <ml:id xml:space="preserve">txsp2</ml:id> <ml:apply> <ml:id xml:space="preserve">rkfixed</ml:id> <ml:sequence> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">txsp1</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">txsp1</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">txsp1</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:real>3</ml:real> </ml:sequence> </ml:apply> </ml:matrix> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">dxsp2dt</ml:id> </ml:sequence> </ml:apply> </ml:localDefine> <ml:return> <ml:apply> <ml:id xml:space="preserve">stack</ml:id> <ml:sequence> <ml:id xml:space="preserve">txsp1</ml:id> <ml:id xml:space="preserve">txsp2</ml:id> </ml:sequence> </ml:apply> </ml:return> </ml:program> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="410" left="294" top="501" width="60" height="12.75" align-x="317.25" align-y="510" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">final</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="331" left="294" top="519" width="129.75" height="16.5" align-x="342" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="f">p</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">txsp</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> <ml:sequence> <ml:id xml:space="preserve">final</ml:id> <ml:real>3</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="333" left="294" top="538.5" width="110.25" height="35.25" align-x="351.75" align-y="558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">prod</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="f">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="29"/> </region> <region region-id="381" left="0" top="576" width="6000" height="6" align-x="0" align-y="576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="362" left="6" top="578.25" width="129" height="12" align-x="6" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Maximize product productivity </text> </region> <region region-id="343" left="6" top="597" width="60.75" height="16.5" align-x="22.5" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.1</ml:real> </ml:sequence> <ml:real>15</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="345" left="6" top="612" width="206.25" height="159" align-x="6" align-y="612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="31"/> <rendering item-idref="32"/> </region> <region region-id="366" left="6" top="770.25" width="86.25" height="12" align-x="6" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Provide initial guess </text> </region> <region region-id="367" left="108" top="771" width="33" height="16.5" align-x="124.5" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="368" left="162" top="771" width="33" height="16.5" align-x="178.5" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f2">t</ml:id> <ml:real>50</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="369" left="210" top="765" width="122.25" height="22.5" align-x="267" align-y="780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.0081466674871140069</ml:real> </result> </ml:eval> </math> <rendering item-idref="35"/> </region> <region region-id="370" left="6" top="789" width="187.5" height="36.75" align-x="36.75" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>9.3179322499810375</ml:real> <ml:real>38.063045545003725</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="371" left="210" top="801" width="90" height="16.5" align-x="267" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.022929483384865081</ml:real> </result> </ml:eval> </math> <rendering item-idref="37"/> </region> <region region-id="380" left="192" top="818.25" width="244.5" height="12" align-x="192" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The productivity in 2 CSTRs in series was 0.027 g/(L<f family="Symbol" charset="2"></f>h) </text> </region> <region region-id="373" left="6" top="836.25" width="25.5" height="12" align-x="6" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> profile </text> </region> <region region-id="374" left="42" top="837" width="79.5" height="16.5" align-x="64.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">txsp</ml:id> <ml:apply> <ml:id xml:space="preserve">txsp</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="f1">t</ml:id> <ml:id xml:space="preserve" subscript="f2">t</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="376" left="144" top="837.75" width="201" height="18" align-x="196.5" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="1" cols="4"> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">p</ml:id> </ml:matrix> <ml:matrix rows="1" cols="4"> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">txsp</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">txsp</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">txsp</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">txsp</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="379" left="6" top="858" width="177" height="129.75" align-x="6" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="40"/> <rendering item-idref="41"/> </region> <region region-id="392" left="6" top="996" width="436.5" height="9.75" align-x="6" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="42" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="43"/> </region> <region region-id="393" left="6" top="1010.25" width="432" height="24" align-x="6" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The product is encoded in a plasmid. Upon cell division, there is a small probability P=0.001 of a plasmid-bearing cell x producing a plasmid-free offspring y. </text> </region> <region region-id="394" left="6" top="1041" width="39" height="12.75" align-x="17.25" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">P</ml:id> <ml:real>0.001</ml:real> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="93" left="6" top="1059" width="186.75" height="16.5" align-x="63.75" align-y="1068" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="dt">dx1</ml:id> <ml:boundVars> <ml:id 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Fermentation time for 5-h induction was longer than that for 10-h induction (~32h versus ~20h). Product concentration for 5-h induction was lower than that for 10-h induction (0.235 g/L versus 0.355 g/L).<sp count="2"/>5-h induction led to a fermentor full of nonproductive plasmid-free cells, the fraction of nonproductive plasmid-free cells being 0.777 for 5-h induction versus 0.022 for 10-h induction. </text> </region> <region region-id="556" left="6" top="2198.25" width="80.25" height="12" align-x="6" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Sub optimal profile </text> </region> <region region-id="557" left="102" top="2199" width="33" height="16.5" align-x="118.5" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" 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