Development of Predictive Growth Model of Vibrio parahaemolyticus Using Mathematical Quantitative Model

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(1)KOREAN J. FOOD SCI. TECHNOL. Vol. 36, No. 2, pp. 349~354 (2004). ©The Korean Society of Food Science and Technology. >' ;ïï&Îj Ï Vibrio parahaemolyticus~ WË .GÎ~ BB ^9·ÁËfÁÖ`1Á¢* ;¯v ·«¦, 1®~£®n*Ó. Development of Predictive Growth Model of Vibrio parahaemolyticus Using Mathematical Quantitative Model Sung-Yang Moon, Tae-Eun Chang, Gun-Jo Woo1, and Il-Shik Shin* Faculty of Marine Bioscience and Technology, Kangnung National University 1 Korea Food and Drug Administration Predictive growth model of Vibrio parahaemolyticus in modified surimi-based imitation crab broth was investigated. Growth curves of V. parahaemolyticus were obtained by measuring cell concentration in culture broth under different conditions (Initial cell level, 1×102, 1×103 and 1×104 colony forming unit (CFU)/mL; temperature, 15, 25 37, and 40oC; pH 6, 7, and 8) and applying them to Gompertz model. Microbial growth indicators, maximum specific growth rate (k), lag time (LT), and generation time (GT), were calculated from Gompertz model. Maximum specific growth rate (k) of V. parahaemolyticus increased with increasing temperature, reaching maximum rate at 37oC. LT and GT were also the shortest at 37oC. pH and initial cell number did not influence k, LT, and GT values significantly (p>0.05). Polynomial model, k = aÁexp (−0.5Á((T− Tmax)/b)2 + ((pH−pHmax)/c)2)), and square root model, k = 0.06 (T-9.55) [1−exp (0.07 (T−49.98))], were developed to express combination effects of temperature and pH under each initial cell number using Gauss-Newton Algorism of Sigma plot 7.0 (SPSS Inc.). Relative coefficients between experimental k and k Predicted by polynomial model were 0.966, 0.979, and 0.965, respectively, at initial cell numbers of 1×102, 1×103, and 1×104 CFU/mL, while that between experimental k and k Predicted by square root model was 0.977. Results revealed growth of V. parahaemolyticus was mainly affected by temperature, and square root model showing effect of temperature was more credible than polynomial model for prediction of V. parahaemolyticus growth. Key words: predictive growth model, Vibrio parahaemolyticus, Gompertz model, polynomial model, square root model, maximum specific growth rate (k). B. . * FÎ>bB HACCP(Hazard Analysis Critical Control Point) Ê~ Bv * ^ê'b ê«> ®b, Ú~ ãÖöê ¢¦ ª¢öB ê«>Ú ÒÏ> ® . $, HACCP Ê~ ê«" z®Ú, "öº " F# & j 7 b ® 7~ b~ Ëæz¢ >' Îj Ï~ .G~¶ ~º & B® ê¯> ® . .Gb(Predictive microbiology)¢ ®ÒÖº ª¢º >' Îj Ï~ ®~ öò¦V B ;, FÛ, &, 6 , ²jö Vræ~ * "; 7öB ÷ ö 5 ¦Nb~ ÿj ;ï'b .G~ BÚj Ï 'b ~ ® . .GböB ÒÏ>º >' Î Bº b~ WËj *~º V Î(1N Î)" b ~ Ãö ~º ~ãº(Nê, pH, >ªW )~ 'Ëj *~º 2N Î Ò ~ «~f V' Ã j «Kb ~ WË æz¢ .G(simulation) > ®º *Î 3N Î W>Ú ® (2,3). .Gb~ 'Ïf ®~ n*W ^Bö ®Ú &Ë 7. ®f '·' &~ 5 V^W ö áæp² *~ î ÷ .O 5 ;j Fæ~º NööB > *' n*W {>Ú¢ò . ¾ î Ú'b ÷öb ö ~ 7ë Ò pª ì B~ Îö ®Ú Bê &;z> ®º º^ . z× vÛ>~ B ~ f * Úö ÖB®~ & * FÛ 7º*~² Úöö V¢ ®~ *' n*W {º B' ^B 6b &v> ® (1). ö ®~ *' n*W {¢. *Corresponding author : Il-Shik Shin, Faculty of Marine Biosience and Technology, Kangnung National University, 123, Jibyun-dong Gangneung-si, Gangwon-do 210-702, Korea Tel: 82-33-640-2346 Fax: 82-33-640-2410 E-mail: [email protected] 349.

(2) ®"²æ B 36 ² B 2 ^ (2004). 350. º~ê j7j Næ~º *bòj &çb ~, ~ Ëj >' Îö ~ ;ï'b .G~ ï& ~V r^ö V~ b ¦Ò»ö j~ Ö ³ . G &Ë~ . æ, *" K, ãj ²º>º b 7«þÁþj &Ú~º B~Bê ÎNW ¸f O»b, &ç ® 7~ ÷ö b~ ;ï' *þê ï&(Quantitative risk assessment)f HACCP Ê~ V 7ö 7 ;{ îÏê J;j * &jÏ~ FÎ>b ;> ® (3-5). FÎWb ~ , ' j 7b B & Úæ ®b, ®ö & '« .V j «K~ Vö ~ êÖj ¯b &ç b ~ Ëæz¢ .GÁjv~ graph * &Ë program ~ BB ê¯>Ú ~ ³ZW(USPA) öBº Pathogen Modeling Program, '~ ³ÚïW(MAFF)öBº Food Micromodelj '' BB~& (6). ¾ programj Ï~z¢ê ® 7~ Î b~ Ãj j*® .G~º ©f ®&Ë~, ê³B program~ BF 5 & ê¯ 7ö ® . Þ, â : ~ Ú~ ãÖ, ®²Ò 7~ >Öb Næ~º j7 ö j~ ¸j öò jî¢. · >Ö&B®~ BB Ú^ z . ¾ ÚÁN~ ¢ 7b >Öb~ ãÖ ßWç *bö ~ J"~ { ¸b, >Öb~ &' ÷öb¢ > ®º Vibrio parahaemolyticus(Ë" j2ÒJ)~ ãÖ, 7ë Ò& j pªì B~ ®Ú ö & &k jº ; . $ WTO ÚB~ Âº ê, B *~ Zö ®ÚBê ®~ *' n*W {¢ * ; ï' ï&»~ ê«" &N ¶òBö & º& 66 Ã &Nö V¢ .Gb~ ê« FÎ >bB ² "Ï> ® . ö öBº >Ö®~ &' 7ë ^, * Ò Ú ç >º¢ Næ~º >Ö&® ²Ú~ b' * º² ;^ ®º V. parahaemolyticusö &~ , B®ö ~ J" ~ WËæz¢ ;ï'b .G~ º >' Î~ BBj * V. ¶ò¢ B~¶, Nê, pHö & V. parahaemolyticus~ WË .GÎj BB~& .. Òò 5 O» " þö ÒÏ "º Vibrio parahaemolyticus ATCC 2210001 «ö F*¶f¯(KCTC)b¦V ª · Aj 3% NaClj Î& brain heart infusion(BHI, Difco Laboratories, Detroit, MI, USA) brothö 15%~ glycerol¢ Î &~ −80oCö ÿÖ&~B þ* modified imitation crab(MIC) brothöB 37oC, 24* *V·~ ÒÏ~& . "~ V· "~ V·f Ëê, ~ WË Îj ËB® ² Úö 'Ï~V * ²Ú Wª(W 7.8%, æO 0.2%, ê>zb 16%, ²ª 2.5%, >ª 73.5%)" FÒ Væ(MIC broth)¢ B~ ÒÏ~& (Table 1). ~ WËö ~º Nê, pH 5 .V > ~ã¶~ 'Ëj rjV *~ ÿÖ "¢ 37oC water bath. Table 1. Composition of modified imitation crab (MIC) broth Ingredients. Concentration. Pepton Disodium phosphate Yeast extract Dextrose Sodium chloride Distilled water. 10 g 2.5 g 10 g 10 g 10 g 1,000 mL. öB /³ ÿ ê, Ò &j MIC broth 50 mLö 7«~ 37oC incubator (Sanyo, Japan)öB 24* *V· ~ & . *V· "¢ 18 mL~ Ò ">(3% NaCl) ¢ Ï 10VO C~ 200 mL~ MIC brothö .V > & 102, 103, 104 CFU/mL& >ê 7«~ Nê 15, 20, 37, 40oC, pH 6, 7, 8öB '' V·~& . ~ WË G; ' þê *~ ã"ö V ~ WËf ¢;* * Ïb plate count agar(PCA, Difco Laboratories, Detroit, MI, USA)¢ ÒÏ~ & ï6 V·»(7)b >¢ G;~ ¾æÚî . Gompertz model~ 'Ï ~ sigmoid ;~ WËj *~V * ÒÏ~º ³>~ f Gompertz& 1825jö ÿ~ ÒÖ"¢ ãþbB B Gompertz function ( 1)(8)j ÒÏ~& b, 1²~ V· þöB *~ ã"ö V 12-16B~ > 8j áîb, 15>j Û þ Ö"8j Gompertz model ö &«~ ~ Gæ~ &Ã³ê ç>(k = BC/e)¢ ~& . LogN = A + CÜexp[−exp{−BÜ(t−M)}]. (1). t: V·*(hr) N(t): V·*ö V >(CFU/mL) A: .V >~ log8(CFU/mL) C: > Ã&ï~ log8(.V >f & >~ N, CFU/mL) M: BG³ê& && >º 6~ * B: * MöB~ BG ³ê(& BG ³ê) ;ï ï& Î~ BB ~ Ãj &~º ~ G æ, & Ã³ê ç>(k)f FêV(LT), ^&*(GT)j þj Û Gompertz Îö ~~ êÂ~&b, & Ã³ê ç>, FêV, ^ &*ö ~º .V >, Nê, pH~ 'Ëj ' ' One-way ANOVA-test¢ ~ Duncan’s multiple range test(9) ¾Ò ï*~ F~W(P<0.05)j SPSS (SPSS Inc., 2000) program (Ver. 10.0)b ¦;~& . $ kö ~º .V>, Nê, pH~ ' &ê¢ square root model" polynomial model ;ïz ~& . Parameter~ Ö;" Î~ ï& Nê 15, 20, 37, 40oC, pH 6, 7, 8 5 .V> 102, 103, 104 CFU/mL~ ~öB Gompertz model¦V & Ã³ê ç>(k)f ~ã "~ &ê¢ «~V *~.

(3) >' ;ïï&Îj Ï Vibrio parahaemolyticus~ WË .GÎ~ BB. 351. Fig. 1. Effects of temperature, pH, and initial cell number on the maximum specific growth rate (k) of V. parahaemolyticus (þ : 15oC, û: 20oC, : 37oC, ø: 40oC). Error bars indicate standard deviations (n=15).. square root model" polynomial modelö '' 'Ï~ SPSS (SPSS Inc., 2000) program(Ver. 10.0)j Ï~ ' ~ parameter¢ Ö;~ Sigma Plot(SPSS Inc., version 7.0)b êz ~&b, ' V· j &«~ B þb¦V & Ã³êç>f Îb¦V & Ã³ ê ç>f~ ç&&ê¢ jv~& .. " polynomial modelj Ï~ BB~&b, BBB ;ï ï & Îö ~~ .GB & Ã³ê ç>¢ Ï~ N ê, pH, .V>ö V V. parahaemolyticus~ WËj Gompertz Î .G, Ã Fb ¾æÚî .. V. parahaemolyticus~ WË .G MIC brothçöB~ V. parahaemolyticus~ Nêf .V>ö V & Ã³ê ç>~ ;ï ï& Îj square root model. ~ã ö 8 ~ G æR~~ æz ' ~ã ºö V ~ Ãj .G~V *~ ~ Ã j &~º Gæ, & Ã³ê ç>(k), FêV. Ö" 5 8. Table 2. Effects of temperature, pH, and initial cell number on lag time and generation time of V. parahaemolyticus pH. Initial cell number (Log CFU/mL). Lag time (hr). Generation time (hr). 15oC. 20oC. 37oC. 40oC. 15oC. 20oC. 37oC. 40oC. 6. 2 3 4. 26.95Û1.45 25.63Û4.56 24.97Û2.10. 5.93Û1.05 7.23Û0.90 5.79Û1.39. 1.35Û0.88 1.66Û0.37 1.56Û0.47. 1.19Û0.26 1.98Û0.68 1.47Û0.46. 3.20Û0.61 3.61Û0.22 3.23Û0.25. 0.90Û0.17 1.00Û0.17 0.96Û0.12. 0.26Û0.04 0.29Û0.02 0.26Û0.03. 0.47Û0.08 0.47Û0.05 0.47Û0.05. 7. 2 3 4. 28.39Û1.53 26.82Û0.56 27.81Û0.99. 6.36Û0.76 7.79Û0.58 6.72Û0.72. 1.84Û1.58 1.61Û0.24 1.42Û0.12. 1.87Û0.16 1.57Û0.26 1.58Û0.47. 2.81Û0.11 3.37Û0.43 2.91Û0.35. 0.86Û0.06 0.95Û0.06 0.91Û0.07. 0.28Û0.04 0.28Û0.02 0.26Û0.04. 0.46Û0.01 0.47Û0.01 0.48Û0.12. 8. 2 3 4. 30.96Û1.01 31.26Û0.85 31.50Û1.18. 6.40Û0.51 6.41Û0.53 6.59Û0.66. 1.66Û1.29 1.89Û0.28 1.63Û0.47. 1.98Û0.26 1.83Û0.26 1.78Û0.47. 2.78Û0.08 2.90Û0.13 2.76Û2.43. 0.88Û0.10 0.93Û0.05 0.88Û0.10. 0.28Û0.04 0.29Û0.02 0.26Û0.01. 0.43Û0.03 0.38Û0.04 0.45Û0.05. Values (meanÛS.D. of 15 times replication) in the same column not sharing a common superscript are significantly different (P<0.05)..

(4) ®"²æ B 36 ² B 2 ^ (2004). 352. Table 3. Parameters of polynomial model for the prediction of the maximum specific growth rate (k) of V. parahaemolyticus at different initial cell numbers Initial level (CFU/mL) 2. 1.0Ü10 1.0Ü103 1.0Ü104 Unification. Parameters a. b. c. Tmax. pHmax. 1.18642 1.11402 1.02322 1.09724. 8.79753 8.93174 9.61076 9.09090. 3.79321 4.26210 9.63489 4.67523. 33.4201 34.2014 34.9752 34.1378. 6.62192 6.89355 4.73513 6.58586. Fig. 2. Comparison of experimental k and predictive k by the response surface model of V. parahaemolyticus at different culture conditions (ù: Experimental k, .: Predictive k).. (LT), ^&*(GT)j Gompertz Î¦V ~& . ~ W Ëj *~º bº æ.ræ &æ~ ; > Ú ®b¾, 7öB ß® Gompertz modelö & & & Ë ôb '" öB '' BBB food micromodel (FMM) and Pathogen Modeling program(PMP) ãÖöê Gompertz modelj ÒÏ~ ® . LogN = A + CÜexp[−exp{−BÜ(t−M)}] k = BC/e GT(hr) = (log 2)e/BC. LT = M − (1/B) V. parahaemolyticus~ Nê, pH, .V>ö V & Ã ³ê ç>~ æz¢ Fig. 1ö ¾æÚî . Nê& ¸jî> &Ã ³êç>º Ã&~ 37oCöB &Ë ¸f 8j ¾æ Úîb, 40oCöBº 6²~º ãËj ¾æÚî . Þ, 37oC, pH 7, .V> 1.0Ü102 CFU/mL ¢r V. parahaemolyticus~ & Ã³êç>º 1.10Û0.16b &Ë ¸f 8j ¾æÚî . ÿ¢ Nê öB pHf .V>¢ Ò þöB º ' Nê 37oC þöB pHf .V>ö V F~.

(5) >' ;ïï&Îj Ï Vibrio parahaemolyticus~ WË .GÎ~ BB. Fig. 3. Comparison of experimental k and predictive k by the square root model of V. parahaemolyticus at different culture conditions (ù: Experimental k , Ç: Predictive k ).. ' N& ìî . ~ NêöBº pH 6" pH 8~ ¢ ¦ þöB .V>ö V &Ã³ê~ F~' N & ¾æÒæò pHf .V>ö V &Ã³êç> æ z~ ¢; ;º æ p~ . º pH¾ .V> . Nê& ~ &Ã³êç>ö æV' 'Ëj ~º © b ¾æÒ . Nê, pH 5 .V>ö V FêV, ^&*~ æz¢ Table 2ö ¾æÚî . FêVº Nê& ¸jî> jæ, 37oCf 40oCöBº jÝ FêV¢ ¾æÚî . ^&* $ Nê& ¸jî> jrb, 37oCöB &Ë f ^& *(ï 0.31Û0.041 hr)j ¾æÚî, 40oCöBº ^Ú r . FêV, ^&*ê & Ã³êf ? pHf .V > Nê~ 'Ëj z Aº ©b ¾æÒ . ~ã ö 8 R&Ã³êç>~ ;ïz Î Nêf pH~ æzö 8 .8>ê ;ïï Î: Nê, pH, .V >ö V V. parahaemolyticus~ WËj ;ï'b .G~V *~ square root model" polynomial modelj Ï~& . ' .V> öB Nêf pH~ ' ' Ëj *~V *B sigmaplot 7.0(SPSS Inc.)~ Gauss-Newton rÒ¾j ÒÏ~ (2)f ?f jF; ²æªCj Û~ Îz~& . f = aÁexp(−0.5(((T−Tmax)/b)2 + ((pH−pHmax)/c)2)). (2). (2)öB a, b, cº ' ~ ç>, Tmaxf pHmaxº & ³êç>& && >º Nêf pH¢ ¾æÞ . ' .V> öB (2)ö ~º ' parameter 8j Table 3ö ¾æ Úîb, .V>ö V &Ã³êç>~ F~' N & ¾æ pbæ, .V>¢ J~æ p, (3)" ?f Nêf pHö & ~¾~ b ¾æÚî . k = 1.10Áexp(−0.5(((T−34.14)/9.09)2 + ((pH−6.59)/4.68)2)). (3). Þ, Fig. 2ö ¾æÂ :f ? pHö V &Ã³êç >(k)~ F~' Nº ìîb(P>0.05), Miles (10)ê V. parahaemolyticus~ &Ã³êç>º pH 6.5öB pH 8.9 º. 353. Fig. 4. Predictive growth curve of V. parahaemolyticus by the polynomial model and square root model (Initial cell number: 1.0Ü103 CFU/mL, Temperature: 37oC, pH 6, ù: Experimental data, : Growth curve by polynomial model, - - -: Growth curve by square root model) .. *öB jv' ¢;~ ~& . ?f pH º*º & ¦ª~ >Ö ®~ Ú~¾ ², îÖ ~ 7'~, N~ ~ · b~ pH º*f ¢~~V r^ö, >Ö ®~ öò $ º &®~ ÷öW ¶ V. parahaemolyticus~ WË .G ÎöB pH¢ æ>öB B~, Ratkowsky (11), Zwietering (12,13) ;~ .V>¢ J~æ pf Nêf & Ã³êç>(k)*~ ç^·Ïj *~º 2N Î square root model~ {ËB ;¢ ÒÏ~ k = b(T − Tmin)[1− exp(c(T − Tmax))] Û~ (4)" ? ;ïz ~&b, Û ê' ªC~¢ Table 4ö ¾æÚî . k = 0.06(T − 9.55)[1 − exp(0.07(T − 49.98))]. (4). Miles (14)f V. parahaemolyticus~ Ã ³êö ~º N êf >ªWö & öB 4 «~~ strain~ V. parahaemolyticus~ G &Ë & Nêº 5.3-8.3oC > , & Nêº 45.3-48oC, ' Nêº 37-39oC ~ & . Þ, þ~ square root modelö ~ V. parahaemolyticus~ G & Nêº 9.5oC, & Nêº 49.9oC f FÒ Ö"¢ ¾æÚî . ;ïï ÎR .G &Ã³êç>(k)f þ~~ j L: Polynomial model( 3)¦V .G V. parahaemolyticus ~ &Ã³êç>f þ~¢ jv Ö"¢ Fig. 2ö w Î ¾æÚî . 1.0Ü102, 1.0Ü103, 1.0Ü104 CFU/mL ~ ' .V> öB þ~f .G~~ ç&ê>º '' 0.966, 0.979, 0.965, .V>¢ J~æ pf Îf ç &ê>& 0.966b ¾æÒ . Þ, square root model~ {Ë B ; (4)¦V .G &Ã³êç>f þ~¢ jv Ö"º Fig. 3" ?b, þ~f .G~~ ç&ê>º 0.977 polynomial model( 3). ² ¸f ç&&ê¢ ¾ æÚî . 2N Î Gompertz Îö ~ V. parahaemolyticus~ WË .G: B .V >, Nê, pH ~öB polynomial model" square root model .G &Ã³êç>(k) ¢ ~ WË Fj *~º Gompertzö 'Ïb, '.

(6) ®"²æ B 36 ² B 2 ^ (2004). 354. Table 4. Parameters of square root model for the prediction of the maximum specific growth rate (k) of V. parahaemolyticus Statistical analysis. Square root model. Tmin Tmax b c. 1). Coefficient. S. E.. t-value. P. S. E. of Estimate. 9.5459 49.980 0.0636 0.0663. 0.3554 0.9566 0.0057 0.0129. 26.8617 52.2472 11.1888 5.1318. <0.0001 <0.0001 <0.0001 <0.0001. 0.0436. 1). Standard error.. öB Gompertz ö ~ ~ WËj .G > ®² > î . .V > 1.0Ü103 CFU/mL, Nê 37oC, pH 6öB~ > þj Û WËF( þ~)" ?f öB~ polynomial model" square root modelö ~ .G WË F(.G~)j jv Ö"º Fig. 4" ? . þj Û &VB ~ WË " .G ;ïï& Îj Ï ~ WËf ¾ ¢~~º © b ¾æÒ . &Ã³êç>ö & ;ï' Î f V. parahaemolyticus~ ~ã ºö & WËj ¶£² Î îVç > ®b, >ªW¾ ¦ z 9f pH º*, Î&b~ ³ê, Ë Ïê~º N2 $º CO2 ³ê¾ *~ ç^·Ï bÒ z' ºö & º&' ¢ ê³ . · ~ã ~öB ; {~² V. parahaemolyticus~ WËj ;ï'b .G > ® j ©b V&B .. º. £. >Ö®öB ^B& >º 7ë V. parahaemolyticus ¢ &çb Nê, pH 5 .V>ö V ~ WË þ Ö "¢ VÆÊz~ ¢ :ûb ~ WËj ;ï' b ï& > ®º >' Îj BB~& . 1.0Ü102, 1.0 Ü103, 1.0Ü104 CFU/mL~ ' .V> öB þ~f . G~~ ç&ê>º '' 0.966, 0.979, 0.965b ¾æÒ . $ , .V>¢ J~æ pf Îf ç&ê>& 0.966b. r" ? ¾æÒ . Polynomial model: k = 1.10Áexp(−0.5(((T−34.14)/9.09)2 + ((pH-6.59)/4.68)2)) ~ Ã æ~ &Ã³êç> kº Nêö æV' 'Ëj A~b, pH 5 .V>ö V F~' Nº ì îbæ (P>0.05), kf Nêf~ &ê square root model ¾æÚî . Square root model: k = 0.06(T − 9.55)[1 − exp(0.07(T − 49.98))] V. parahaemolyticus~ ãÖ, square root modelö ~ þ~ f .G~~ ç&ê>º 0.977 polynomial model ¸f ' ÏWj ¾æÚî .. 6Ò~ º 2000jê >Öß;BBÒë~ ¢~b · >Ö¦~ j æöö ~ >¯B Ö", ö p 6Òãî .. ^. ò. 1. Ross T, McMeekin TA. Predictive microbiology. Int. J. Food Microbiol. 23: 241-264 (1994) 2. Coleman ME, Marks HM. Qualitative and quantitative risk assessment. Food Control 10: 289-297 (1999) 3. Baker DA. Application of modelling in HACCP plan development. Int. J. Food Microbiol. 25: 251-261 (1995) 4. Notermans S, Gallhoff G, Zwitering MH, Mead GC. The HACCP concept: specification of criteria using quantitative risk assesment. Food Microbiol. 12: 81-90 (1995) 5. Ross T, McMeekin TA. Predictive microbiology. Int. J. Food Microbiol. 23: 241-264 (1994) 6. Yano N. Predictive Microbiology and its application in food industry. Jpn. J. Food Microbiol. 15: 81-87 (1998) 7. AOAC. Official Methods of Analysis. Method 940.36. Association of Official Analytical Chemists, Arlington, VA, USA (2000) 8. Ratkowsky DA, Ross T. Modelling the bacterial growth/no growth interface. Lett. Appl. Microbiol. 20: 29-33 (1995) 9. Duncan DB. Multiple-range and multiple F test. Biometrics 11: 1-42(1955) 10. Miles DW. Predicting the growth of Vibrio parahaemolyticus. BS thesis, University of Tasmania, Tasmania, AU (1994) 11. Ratkowsky DA, Lowry RK, McMeekin TA, Stokes AN, Chandler RE. Model for bacterial culture growth rate through the entire biokinetic temperature range, J. Bacteriol. 154: 1222-1226 (1983) 12. Zwietering MH, de Koos JT, Hasenack BE, de Wit JC, van 'T Riet K. Modeling of bacterial growth as a function of temperature. Appl. Environ. Milcobiol. 57: 1094-1101 (1991) 13. Zwietering MH, Cuppers HGAH, de Wit JC, van 'T Riet K. Evaluation of data transformations and validation of a model for the effect of temperature on bacterial growth. Appl. Environ. Microbiol. 60: 195-203 (1994) 14. Miles DW, Thomas R, Olley J, Thomas A, McMeekin TA. Development and evaluation of a predictive model for the effect of temperature and water activity on the growth rate of Vibrio parahaemolyticus, Int. J. Food Microbiol. 38: 133-142 (1997) (2003j 12ú 24¢ 7>; 2004j 3ú 30¢ j).

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