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RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S H T @@@@ #" `/|3   b* -DDD  H T @@@@ #" `/u 13    d* -DDD  H Tp @@@@ #" `/f5H3   d* -DDD z H NA!nF ?clouds_3#" @@ fy380___PPT10.ݿ 8,  Clouds  0 910L(  L  L Zܽ @@@@ ?#" `4D{   T Click to edit Master title style! ! L N @@@@ "- U?E+   W#Click to edit Master subtitle style$ $ L T @@@@ #" `/|3   b* -DDD  L T @@@@ #" `/u 13    d* -DDD  L T @@@@ #" `/f5H3   d* -DDD z L NA!nF ?clouds_3#" @@ fy380___PPT10.ݿ 8,K- 0 b,Z, 5+(  ^  6Ԕ35M1*^  6ԔQ  ^   6Ԕ3   0 LS" 7?    P  C (Aoxcrest7v ^  C 6Amicrosoftcrest7?I   0 ~ ve  ( 2_   0 ]I& G2. The Model In his paper in 1952 [4], Turing suggested that chemicals can react and diffuse to produce patterns. Therefore, the formation of patterns in an organism can be modelled by a series of reaction-diffusion equations for the interacting species. If, on the removal of the diffusion terms, the species tend towards a linearly stable state, then, under certain conditions, patterns can evolve when the diffusion is re-introduced. A non-dimensionalised model for the interaction of two species, u and v, was proposed by Schnakenberg [3] in 1979. In one spatial dimension, it ishH(2 $L,D .`   c $A  ??&Oi,    0 5,E3 ) represents the relative strength of the reaction terms d is the ratio of diffusion coefficients a and b are different ratios of the positive rate constants Neumann boundary conditions are applied on the domain [0,1].48 g2 g2^  6ԔQ 5N1#^  6ԔQ 2fI)  0 50  $4. Conditions needed for pattern formation After analysis it was found that the following conditions needed to be satisfied for diffusion driven instability to take place, i.e. for pattern formation to occur.8+ 2(2+$`  c $A  ??9!U+)  /  0   IoO 1. Introduction Pattern formation is an important part of developmental biology. In the formation of an organism, cells differentiate, in the biological sense, according to where they are in the spatial organisation [2]. We have been looking at methods of modelling pattern formation, and how, by altering the parameters and initial conditions of the model, different pattern types can form.,(2$x  0  b0 13. Solution The solution of the PDEs has the form8 2&(2 $& `  c $A ??7M'   0 gb0 Iwhere U, V are constants. Using the boundary conditions, it is shown that:J(2@`  c $A ??k#P'/ Z  0   b0t hwhere n is an integer. k is the mode, and determines how many stripes form. e.g. k = 1 gives one stripe.vi(29  0Y~ @3H 5. Results Analysis showed that varying g had the greatest effect upon the type of pattern formed with this model. By choosing g appropriately, different modes, k, could be isolated separately, allowing for an increasing number of stripes to form. For examplej 2(2 $ya h 3H  #"3H ' <|y?D=H Q4D  & <Hy?]@=D Q3D  % <œ?<=]@ Q2D  $ <?7=< Q1D  # <a?3=7 ]Number of StripesD  " <H?DH= S450D  ! <A?]@D= S250D    <ky?<]@= S115D   <`y?7<= R30D   <@y?37= RgD ZB + s *jJ ?33ZB ( s *jJ ?3HZB 0 s *jJ ?HHZB * s *jJ ?3HTB r c $jJ ?77TB s c $jJ ?<<TB t c $jJ ?]@]@TB u c $jJ ?DDTB v c $jJ ?3=H=H  C A450F=>H&A  08|@3 I The other parameters took the values: a = 0.1, b = 1, d = 10. These results agreed with computer simulations run on MATLAB, using random oscillatory initial conditions.(2H  C A 115F3j>'  0 '5=( 0(2  0|&~4=( ~g = 115. Two intersections at steady state, therefore 2 stripes@(2@  0 &>G( g = 450. Four intersections at steady state, therefore 4 stripes.B(2B^  6Ԕ)2fI3  0 B*3 I3 d6. Future work Analyse the model in two spatial dimensions, to establish 2D patterns, such as spots. Look at other models, such as a Gierer-Meinhardt [1] system Examine sequential pattern formation due to stimulation at one boundary.: 2(2$V^  6Ԕ,bM13P  0( ,03 References [1] A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12:30-39, 1972. [2] J. Murray. Mathematical Biology. Springer-Verlag, Berlin, Heidelberg, 2nd edition, 1989. [3] J. Schnakenberg. Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81:389-400, 1979. [4] A. Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. 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Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0/ 00DAriallack0LL|pԖ0ԖDWingdingsk0LL|pԖ0Ԗ DArial Black0LL|pԖ0Ԗ"0DSymbollack0LL|pԖ0Ԗ@ .  @n?" dd@  @@`` TB..R$d!6sT^IAzR$is߸8l`eDAzR$X'mVD8 D2$ZR\92$I#ayn`q@R$Xns ]P!2$7aga2$5@`_N^údR$/&7!Fz 0AA@8  @u . 0 -- 6o-(  ^  6Ԕ5M1)^  6ԔQ ^   6ԔI3   0.LS" 7?   P  C (Aoxcrest7v ^  C 6Amicrosoftcrest7?I   0p1~ ve  ( 2I   0T4v& C2. The Model In his paper in 1952 [4], Turing suggested that chemicals can react and diffuse to produce patterns. Therefore, the formation of patterns in an organQvGHtGDEŃfYmiXO Ygt`Yg5^NV+kcW[; ^EQ>2^Vd^ b!Sg^rwa !W*ۭ?fnu=.J8aMI7O4Owѿk]sk1y:n$] WިMZ#i.}Ee iLy|3?%DUx=˜d^[Iߐ KNe^{M')ɀo=2j#f?@v))ڔg).FUj?I+׮罤Ow^0]M*6Ì( Y?3M,/Otion 3.0 Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0/ 00DAriallack0LLrԖ0Ԗ"DWingdingsk0LLrԖ0Ԗ DArial Black0LLrԖ0Ԗ"0DSymbollack0LLrԖ0Ԗ@ .  @n?" dd@  @@`` TB..R$d!6sT^IAzR$is߸8l`eDAzR$X'mVD8 D2$ZR\92$I#ayn`q@R$Xns ]P!2$7aga2$5@`_N^údR$/&7!Fz 0AA@8  @t . 0 -- 6n-(   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ ^  6Ԕ5M1)^  6ԔQ ^   6ԔI3   0eLS" 7?   P  C (Aoxcrest7v ^  C 6Amicrosoftcrest7?I   0|i~ ve  ( 2I   0`lv& C2. The Model In his paper in 1952 [4], Turing suggested that chemicals can react and diffuse to produce patterns. Therefore, the formation of patterns in an organism can be modelled by a series of reaction-diffusion equations for the interacting species. If, on the removal of the diffusion terms, the species tend towards a linearly stable state, then, under certain conditions, patterns can evolve when the diffusion is re-introduced. A non-dimensional model for the interaction of two species, u and v, was proposed by Schnakenberg [3] in 1979. In one spatial dimensiPowerPoint Document(<DocumentSummaryInformation8 Root EntrydO)@xʤPictures <Current User,SummaryInformation(P  ( !"#$%&'*+,-./0123456789:;<=>) _JUPPJUPP՜.+,0    kCustomUniversity of Oxford<+ Arial Wingdings Arial BlackSymbolCloudsMicrosoft Equation 3.0Reaction-Diffusion on, it ishD(2 $L  .`   c $A ??'Oi],   0z,I3 < represents the relative strength of the reaction terms, d is the ratio of diffusion coefficients, a and b are different ratios of the positive rate constants. Neumann boundary conditions are applied on the domain [0,1].D9 g2 g2^  6ԔQ 5N1^  6ԔQ 2fI)  050  "4. Conditions needed for pattern formation After analysis it was found that the following conditions need to be satisfied for diffusion driven instability to take place, i.e. for pattern formation to occur.8+ 2(2+$`  c $A  ?? *,!)  /  0 IoO 1. Introduction Pattern formation is an important part of developmental biology. In the formation of an organism, cells differentiate, in the biological sense, according to where they are in the spatial organisation [2]. We have been looking at methods of modelling pattern formation, and how, by altering the parameters and initial conditions of the model, different pattern types can form.,(2$x+  0 b0 U3. Solution The solution of the linearised system about the steady state has the form8 2J(2 $J ,`  c $A ??'y   0gb0 Iwhere U, V are constants. Using the boundary conditions, it is shown that:J(2@`  c $A ??k#P'/ Z  0 b0t hwhere n is an integer. k is the mode, and determines how many stripes form. e.g. k = 1 gives one stripe.vi(29  0\~ @3H 5. Results Analysis showed that varying g had the greatest effect upon the type of pattern formed with this model. By choosing g appropriately, different modes, k, could be isolated separately, allowing for an increasing number of stripes to form. For examplej 2(2 $ya h 3H  #"3H\ ' <`?D=H Q4D  & <?]@=D Q3D  % <?<=]@ Q2D  $ <p?7=< Q1D  # <?3=7 ]Number of StripesD  " <?DH= S450D  ! <h?]@D= S250D    <?<]@= S115D   <?7<= R30D   <H ?37= RgD ZB + s *jJ ?33ZB ( s *jJ ?3HZB 0 s *jJ ?HHZB * s *jJ ?3HTB r c $jJ ?77TB s c $jJ ?<<TB t c $jJ ?]@]@TB u c $jJ ?DDTB v c $jJ ?3=H=H  C A450F=>H&C  03H The other parameters took the values: a = 0.1, b = 1, d = 10. These results agreed with computer simulations run using MATLAB, with random oscillatory initial conditions.(2H  C A 115F3j>'  0H'5=( 0(2  0&~4=( g = 115. Two intersections at steady state, therefore 2 stripes.A(2A  0D"&>G( g = 450. Four intersections at steady state, therefore 4 stripes.B(2B^  6Ԕ)2fI3  0l'B*3 I3 d6. Future work Analyse the model in two spatial dimensions, to establish 2D patterns, such as spots. Look at other models, such as a Gierer-Meinhardt [1] system Examine sequential pattern formation due to stimulation at one boundary.: 2(2$V^  6Ԕp*5M10f  080p*bj00 References [1] A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12:30-39, 1972. [2] J. Murray. Mathematical Biology. Springer-Verlag, Berlin, Heidelberg, 2nd edition, 1989. [3] J. Schnakenberg. Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81:389-400, 1979. [4] A. Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. B., 237:37-72, 1952. 2(2 J  'bFt , @0 By^  6Ԕ7vI ;  68]Ԕ15M13 cAcknowledgements. This research was financially supported by a research studentship from Microsoft.: 2R(2Rc H  0!nF ? 33___PPT10i. /f+D=' ]= @B + &xXilTU>7鴝P.V& T@Ŷ1AZeB-#TCd?$!6.  Ը@l?H6$ZϹo}S:grysr?O3ࠧ@lPw Bx7088H +F% ЀMt¨4֙?u9~Fw7>(lӧ^ PwM6eҀ3m1#O2Z +>4>TRYE0"tcLo3xơ ^s3F? hS|_uĀDtL|lgA&+_# 2Ŀ9zcf1/S&79s"ß32Lt21=O^1ψk~2a r.+=)ȅ!OE8]č+4CB>ΫcqKzn։aNi]Ig= ?ek~\s!/?F,~&ՊRM>E y.)S b-Y2v5Łq>:ٜLCg:lC߅趹aڕ*W.:g B5lElDbcK4kl++piNGS(>v}&{[43M {Uh+)#H;g:ZVv.ַ 2 uDEHt@O ?wbo-ъcvB Y)wٮmYTh&Lc43PsBeiv%y^1xq14i4uVy1;IZ?㝺uxϻ>8>_ݾUGZŰjE)oK L`_* C9+^&c:m vxRfK^bJC*c;/e%H`Q3hbGv<+>qR&^I6u4G1R" c=eʑێQ>W{hX5MYP2zbvҜsXOLucTH&C  0@3H The other parameters took the values: a = 0.1, b = 1, d = 10. These results agreed with computer simulations run using MATLAB, with random oscillatory initial conditions.(2H  C A 115F3j>'  0'5=( 0(2  0&~4=( g = 115. Two intersections at steady state, therefore 2 stripes.A(2A  0@&>G( g = 450. Four intersections at steady state, therefore 4 stripes.B(2B^  6Ԕ)2fI3  0dB*3 I3 e6. Future work Analyse the model in two spatial dimensions, to establish 2D patterns, such as spots. Look at other models, such as a Gierer-Meinhardt [1] system. Examine sequential pattern formation due to stimulation at one boundary.: 2(2$W^  6Ԕp*5M10f  0p*bj00 References [1] A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12:30-39, 1972. [2] J. Murray. Mathematical Biology. Springer-Verlag, Berlin, Heidelberg, 2nd edition, 1989. [3] J. Schnakenberg. Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81:389-400, 1979. [4] A. Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. B., 237:37-72, 1952. 2(2 J  'bFt , @0 By^  6Ԕ7vI ;  6Ԕ15M13 cAcknowledgements. This research was financially supported by a research studentship from Microsoft.: 2R(2Rc H  0!nF ? 33___PPT10i. /f+D=' = @B +r01