Informacije

Kako mogu usporediti krivulje rasta bakterija?


Imam krivulje rasta bakterija za 5 različitih temperaturnih uvjeta i one su stvorene pomoću %propusnosti crvenog svjetla. Kako mogu uporediti ove krive rasta? Trebam li izračunati stopu rasta i ako da, kako to učiniti?


1) Kriva rasta bakterija:

Kada bakterijama osiguramo povoljan medij za redovnu podjelu i povećanje broja, dobivamo ono što se naziva "krivulja rasta", mjereći koncentraciju stanica s vremenom kako vrijeme prolazi.

2) Faze:

A) Prva faza se naziva "faza zaostajanja", u ovoj fazi bakterije se prilagođavaju novom mediju, pripremajući enzime itd., Za početak diobe. Stoga se ne primjećuje mali rast ili nikakav rast.

B) Druga faza je "faza eksponencijalnog rasta", bakterijske ćelije su dobro prilagođene svom mediju i spremne za postizanje punog potencijala, ovdje se događa "maksimalna podjela i rast".

C) Treća faza je "stacionarna faza", bakterije više ne mogu rasti niti se hraniti, ponestaje im hranjivih tvari, a rast je opet stacionaran/nula. Može trajati dugo vremena. (Rast ćelija = ćelijska smrt, nema stvarnog povećanja broja ćelija = stacionarno)

D) Konačna faza rasta je "faza smrti", nema dostupne hrane i toksični metaboliti su se nakupljali tokom stacionarne faze, dolazi do smrti (smrt ovdje znači da sve veći broj bakterijskih stanica umire, za razliku od stacionarne faze u kojoj broj mrtvih stanica jednak je broju održivih stanica), nakon smrti ćelije (koja se sada događa brže od rasta ćelija), koncentracija ćelija počinje opadati. Međutim, to zapravo ne doseže nulu i bakterije bi se mogle ponovno regenerirati kad uvjeti ponovo budu zdravi.

3) Bilježenje rasta bakterija pomoću propusnosti ili spektrofotometra:

Kako se broj bakterijskih stanica povećava, zamućenost njihovog medija raste (ovisi o koncentraciji stanica), a upadno svjetlo (ako ispuštate snop svjetlosti preko medija) sve se više odbija od prijemnog detektora (radi otkrivanja intenzitet svjetlosti), što čini spektrofotometar pouzdanom i preciznom metodom za mjerenje rasta ćelija (proporcionalno intenzitetu otkrivene svjetlosti).

4) Kako uporediti krive rasta?

Pogled na krivulju rasta dat će neku ideju o stopi rasta u odnosu na standardnu ​​krivulju rasta ili neke druge dostupne krivulje rasta (usporedite faze i njihovo trajanje, eksponencijalnu fazu itd.). Međutim, postoji način da se iz krivulje rasta izračuna krivulja rasta i vrijeme stvaranja (vrijeme udvostručenja).

Razlika u koncentraciji ćelija/razlika u vremenu = konstanta brzine rasta. Ili dN/dt = kN ili možete koristiti ovu naprednu jednadžbu: log10N2 - dnevnik10N1 = k (t2 - t1)/2.303

Jednostavno iscrtajte bilo koje dvije slučajne tačke vremena na vašoj krivulji: prva tačka je t1, druga tačka je t2, N1 je broj ćelija (koncentracija ćelija) pri t1, N2 je koncentracija ćelija pri t2.

Ili jednostavno možete koristiti dN/dt = (N1 - N2)/(t1 - t2) i usporedite rezultirajući k između krivulja.

Nadam se da je ovo odgovor na vaše pitanje, imajte na umu da ja ni u kom slučaju nisam stručnjak za tu temu, pa možete potražiti dalje ili dobiti savjet stručnjaka ako je potrebno.

Reference:

Textbookofbacteriology.net

Miller-lab.net/MillerLab

biotek.com/

Academic.pgcc.edu

Web.mst.edu


Izračunavanje maksimalnih stopa rasta

Maksimalno stope rasta su zaista važne za usporedbu pri procjeni razlika između različitih krivulja rasta. Što je veća vrijednost, najbrža je eksponencijalna faza. Da biste izračunali brzinu rasta svake krivulje, morate imati barem mogućnosti:

Najbrži i najjednostavniji način je odabir dvije točke iz eksponencijalna fazi i koristite sljedeću formulu:

$ mu_ {max} = frac {ln (X_2) - ln (X_1)} {t_2 - t_1} $

gdje je $ X_i $ vaša mjera gustoće ćelija (npr. suha težina, optička gustoća, itd.) u trenutku $ t_i $, a $ mu_ {max} $ je maksimalna stopa rasta.

Drugi način procjene $ mu_ {max} $ je korištenje tehnika modeliranja. Klasičan model za rast bakterija je Gompertzov model:

$ y (t) = A cdot exp lijevo [ - exp lijevo (1 + frac { mu cdot e} {A} ( lambda - t) desno) desno) $

Gdje je $ y $ mjera gustoće ćelije, $ A $ asimptotska vrijednost gustoće ćelije i $ lambda $ je trajanje faze kašnjenja. Postoje alati koji vam mogu pomoći da prilagodite svoje obline ovoj vrsti modela, poput grofit paketa za R (besplatan je!). Ovaj paket može vam dati procjene $ A $, $ lambda $ i $ mu_ {max} $, zajedno sa odgovarajućim intervalima povjerenja (pogledajte sliku ispod).

Poređenje kriva rasta

Kada se upoređuju krivulje rasta, ključne vrijednosti za usporedbu su: $ mu_ {max} $ maksimalna stopa rasta, $ A $ asimptotska vrijednost i $ lambda $ trajanje faze kašnjenja. Ne treba zanemariti jedno od ostalih. Niske $ mu_ {max} $, niske $ A $ ili visoke $ lambda $ znak su mogućeg bakterijskog stresa.


Izvor slike: Kahm, M., Hasenbrink, G., Lichtenberg-Fraté, H., Ludwig, J., Kschischo, M. (2010) grofit: Uklapanje krivulja biološkog rasta s R, Journal of Statistical Software (33)7


Faktori rasta bakterija

Bakterijski faktori rasta prvenstveno uključuju temperaturu, pH, koncentraciju soli, izvor svjetlosti, nutritivne i plinovite potrebe itd. Mikroorganizmi žive na različitim prirodnim staništima kao što su atmosfera, hidrosfera i litosfera.

Ekološki i nutritivni čimbenici mogu pogodovati ili ograničiti rast mikroorganizama. Neki mikrobi opstaju u ekstremnim uvjetima ili promjenjivoj okolini, dok rijetki ne mogu uspjeti. Faktori rasta bakterija odlučuju o povećanju veličine i mase ćelije bakterije.

U ovom ćemo članku raspravljati o faktorima rasta bakterija kao što su temperatura, pH, koncentracija kisika, ugljični dioksid, svjetlost i osmotski tlak.

Sadržaj: Faktori rasta bakterija

Temperature

To je jedan od najvažnijih faktora koji odlučuje o stopi razmnožavanja mikroorganizama. Temperatura može biti minimalna, optimalna i maksimalna.

  • Minimalna temperatura: ispod koje ne dolazi do rasta.
  • Optimalna temperatura: Na kojoj dolazi do najbržeg rasta.
  • Maksimalna temperatura: Iznad koje ne dolazi do rasta.

Kardinalne temperature za rast mikroba: Mikroorganizmi pripadaju različitim grupama prema optimalnoj temperaturi za podjelu i metabolizam.

Psihrofili: Mikroorganizmi koji pripadaju ovoj grupi su „Ljubav prema hladnoći”, A mogu rasti na temperaturi od nula stepeni Celzijusa. Uključuje dvije podgrupe:

  • Pravi psihrofili: Optimalni rast postižu na 15 stepeni Celzijusa ili ispod, ali su osjetljivi na temperature iznad 20 stepeni Celzijusa. Sveprisutan je na sjevernom polu i dubini okeana.
  • Psihrotrofi: Psihrotrofi postižu optimalan rast između 20-30 stepeni Celzijusa. One su sveprisutne prirode u odnosu na psihrofile.

Mezofili: Mikroorganizmi koji pripadaju ovoj grupi zahtijevaju temperaturu u srednji domet. Najbolje rastu na normalnoj temperaturi u rasponu od 25-40 stepeni Celzijusa. Optimalna temperatura za mezofilne bakterije je 37 stepeni Celzijusa. Uključuje komenzale, saprofite, biljne parazite itd.

Termofili: Mikroorganizmi koji pripadaju ovoj grupi su „Ljubitelj toplote”, Gdje oni pokazuju optimalan rast na temperaturi između 50-60 stepeni Celzijusa. Mnogi termofilni organizmi ne mogu rasti ispod 45 stepeni Celzijusa.

Bakterije koje pripadaju ovoj grupi mogu se prilagoditi životu u teškim okruženjima poput vrućih izvora, sunca obasjanog tla, gomila komposta itd. Neke termofilne bakterije također posjeduju endospore otporne na toplinu.

Termofili ili hipertermofili predstavljaju drugu grupu koja najbolje raste na temperaturi od 80 stepeni Celzijusa ili više. Termofilne bakterije poput Archaebacteria, Thermus aquaticus prevladavaju u vulkanskim i okeanskim otvorima.

Bakterije su osjetljive na različite pH vrijednosti. Bakterijske vrste koje rastu na rasponu pH iznad ili ispod željene vrijednosti ne bi preživjele. No, bakterije pri optimalnom pH -u pokazuju najbolji rast pri gotovo neutralnom pH, tj. 6.5-7.5.

Kiseli pH obično ograničava rast mikroba i često se koristi u metodama konzerviranja hrane. Osnovni pH također inhibira rast mikroba. Za stabilizaciju pH općenito se koriste puferi. Stoga se, ovisno o rasponu pH, organizmi mogu grupirati u tri kategorije:

  1. Acidophiles: Organizmi koji pripadaju ovoj grupi su „Acid loving”Rastu u vrlo niskom rasponu pH (između 0,1-5,4). Bacillus sp, Micrococcus sp, Sulphobolus sp. itd., spada u kategoriju acidofilnih organizama.
  2. Neutrofili: Organizmi koji pripadaju ovoj grupi rastu na normalnom rasponu pH (između 5,4-8,5). Uključuje većinu ljudskih kommenzala, jer članovi pripadaju porodici Enterobacteriaceae.
  3. Alkalofili: Organizmi koji pripadaju ovoj grupi su „Ljubljenje alkalija”Rastu na alkalnom rasponu pH (između 7-12). Vibrio cholera, Alkaligenes sp, Agrobacterium sp itd., spada u kategoriju alkalofilnih organizama.

Mikroorganizmi se mogu prilagoditi promjenama pH okoliša održavanjem unutrašnjeg pH koji je blizu neutralnosti. Neke bakterije također sintetiziraju proteine ​​šoka kao odgovor na pH. PH medija također određuje koji će put metabolizma stupiti u akciju.

Koncentracija kisika

Bakterijske vrste koje koriste molekularni kisik (O2) proizvode više energije iz hranjivih tvari nego anaerobi. Kisik funkcionira kao terminalni akceptor elektrona za transportni lanac elektrona tijekom aerobnog disanja. Na osnovu koncentracije kiseonika, mikroorganizmi se mogu svrstati u sljedeće tipove:

  1. Obavezni aerobi uključuju mikroorganizme poput Pseudomonas sp, uobičajeni bolnički patogeni itd., kojima je za preživljavanje potreban kisik.
  2. Fakultativni anaerobi uključiti E. coli, Staphylococcus sp, itd., koji koristi kisik i može rasti i bez njegovog prisustva.
  3. Obavezni anaerobi uključiti Clostridium sp, Pasteurianum sp itd., koji ne mogu koristiti kisik, a oštećeni su prisutnošću toksičnih oblika kisika.
  4. Aerotolerantni anaerobi uključiti Lactobacillus sp, Enterococcus faecalis itd., koji mogu koristiti kisik, ali mogu tolerirati njegovo prisustvo.
  5. Mikroaerofili uključiti Campylobacter species kojima je potreban kisik, ali u niskim koncentracijama i osjetljivi na otrovne oblike kisika.

Osnove različite osjetljivosti na kisik

Kisik može lako smanjiti toksične proizvode poput singletnog kisika i vodikovog peroksida. Singletni kisik je izuzetno reaktivan oblik koji se uglavnom javlja u fagocitnim stanicama. Jon vodikovog peroksida također je otrovan, koji se može razgraditi pomoću dva enzima, odnosno katalaze i peroksidaze.

Slobodni radikal superoksida je izuzetno otrovan i reaktivan oblik kisika. Organizmi koji rastu na ovoj temperaturi uključuju aerobe, fakultativne anaerobe i aerotolerante, ali ne i anaerobe ili mikroaerofile.

Ugljen-dioksid

Odnosi se na grupu bakterija koja za svoj rast koristi veću količinu CO2 kapnofilne bakterije (H. influenzae, Brucella abortus, itd.). Za optimalan rast potrebno im je prisustvo 5-10% CO2 i 15% O2. U staklenci za svijeće može se postići 3% CO2. Capnofili se nalaze u normalnoj flori nekih preživara.

Light

To je još jedan faktor koji utječe na rast bakterija, a one bakterije koje koriste izvor svjetlosti mogu se klasificirati kao:

Fototrofi: Odnosi se na skupinu bakterija koja energiju crpi hvatanjem fotona uglavnom iz sunčevu svetlost. Fototrofi se mogu klasifikovati u autotrofe (fiksni ugljenik) ili heterotrofe (koristi ugljenik). Primjeri: Rhodobacter capsulatus, Chromatium, Klorobij itd.

Hemotrofi: Odnosi se na skupinu bakterija koje dobivaju energiju oksidirajući elektrone prvenstveno iz hemijski izvori. Organski (hemoorganotrofi) ili anorganski (hemolithotrofi) su dvije uobičajene vrste. Oni prevladavaju na dnu oceana gdje sunčeva svjetlost ne može doseći.

Osmotski pritisak

Mikrobi za svoj rast zahtijevaju minerale ili hranjive tvari, koje se mogu dobiti iz okolne vode. Osmotski tlak i koncentracija soli otopine mogu utjecati na rast bakterija. Bakterijska stanična stijenka daje mehaničku snagu koja omogućava bakterijama da izdrže izmjene osmotskog tlaka.

Osmofilne bakterije zahtijevaju visok osmotski tlak. Kada se bakterijska stanica podvrgne hipertoničnoj otopini, to može uzrokovati osmotsko uklanjanje vode, što rezultira plazmoliza ili osmotsko skupljanje protoplazme.

Nasuprot tome, kada se bakterijska ćelija podvrgne destiliranoj vodi iz velike koncentracije, to može uzrokovati pretjerano upijanje vode što rezultira plazmoptiza ili pucanje ćelija.


Kriva rasta bakterija: 4 faze

Nakon inokulacije u sterilni hranjivi medij, bakterija prvo pod & shygoom prolazi period aklimatizacije. U to vrijeme se sintetiziraju potrebni enzimi i intermedijarni metaboliti, pa bakterija i širim dostižu kritičnu fazu prije multi & shyplikacije, u ovoj fazi dolazi do umnožavanja.

Trajanje faze kašnjenja ovisi o vrsti bakterije, kvaliteti podloge za uzgoj, veličini inokuluma i nekoliko faktora okoline, poput CO2, temperatura, pH itd. Prosječno vrijeme faze kašnjenja je 2 sata, iako varira od vrste do vrste (1-4 sata).

2. Faza dnevnika ili eksponencijalna faza:

U ovoj fazi bakterije prolaze diobu stanica i njihova se populacija (broj) eksponencijalno povećava logaritamskom brzinom. Broj održivih brojeva, kada se iscrtava prema vremenu, daje ravnu liniju nagnute mode. Prosječno vrijeme log faze je 8 sati, iako varira kod različitih vrsta.

3. Stacionarna faza:

U ovoj fazi rast, odnosno dioba stanica, gotovo prestaje uslijed iscrpljivanja hranjivih tvari, kao i nakupljanja toksičnih proizvoda. U ovoj fazi ćelijska smrt počinje sporim tempom i kompenzira se stvaranjem nove ćelije diobom.

Ukupan broj ćelija sporo se povećava, ali održiv broj ostaje gotovo konstantan. Trajanje ove faze je promjenjivo i kreće se od nekoliko dana do nekoliko sati. U ovoj fazi nastaju sekundarni metaboliti poput antibiotika, toksina itd.

4. Faza opadanja:

U fazi opadanja, ukupan broj stanica ostaje konstantan, ali se broj održivih stanica postupno smanjuje zbog iscrpljivanja hranjivih tvari i nakupljanja otrovnih proizvoda. U nekim slučajevima nekoliko ćelija ostaje održivo dugo vremena, čak i nakon smrti većine stanica. Ove održive ćelije vjerojatno rastu korištenjem hranjivih tvari oslobođenih iz mrtvih stanica.

Ćelije postižu najveću veličinu na kraju faze kašnjenja i postaju manje u log fazi (eksponencijalna faza). Kod vrsta koje stvaraju spore sporulacija se javlja na kraju log faze (eksponencijalna faza) ili u ranom dijelu stacionarne faze.


Princip:


Povećanje veličine i ćelijske mase tokom razvoja organizma naziva se rast. To su jedinstvene karakteristike svih organizama. Organizam mora zahtijevati određene osnovne parametre za proizvodnju energije i staničnu biosintezu. Na rast organizma utiču fizički i nutritivni faktori. Fizički faktori uključuju pH, temperaturu, osmotski tlak, hidrostatički tlak i sadržaj vlage u mediju u kojem organizam raste. Nutritivni faktori uključuju količinu ugljika, dušika, sumpora, fosfora i drugih elemenata u tragovima koji se nalaze u mediju za rast. Bakterije su jednostanični (jednoćelijski) organizmi. Kad bakterije dosegnu određenu veličinu, dijele se binarnom fisijom, u kojoj se jedna ćelija dijeli na dvije, dvije na četiri i nastavljaju proces na geometrijski način. Tada se zna da je bakterija u aktivno rastućoj fazi. Da bi se proučila populacija rasta bakterija, vitalne ćelije bakterije treba inokulirati u sterilnu juhu i inkubirati u optimalnim uvjetima rasta. Bakterija počinje koristiti komponente medija te će povećati svoju veličinu i staničnu masu. Dinamika rasta bakterija može se proučavati iscrtavanjem rasta ćelija (apsorbancije) u odnosu na vrijeme inkubacije ili zapis broja ćelija u odnosu na vrijeme. Tako dobivena krivulja je sigmoidna krivulja i poznata je kao standardna krivulja rasta. Povećanje stanične mase organizma mjeri se pomoću spektrofotometra. Spektrofotometar mjeri zamućenost ili optičku gustoću koja je mjera količine svjetlosti koju apsorbira bakterijska suspenzija. Stupanj zamućenosti u kulturi bujona izravno je povezan s brojem prisutnih mikroorganizama, bilo živih ili mrtvih stanica, te je zgodna i brza metoda mjerenja brzine rasta stanica u organizmu. Tako povećanje zamućenosti medijuma za bujon ukazuje na povećanje mikrobne ćelijske mase (slika 1). Količina propuštene svjetlosti kroz zamućenu juhu smanjuje se s naknadnim povećanjem vrijednosti upijanja.

Slika 1: Očitavanje apsorbancije bakterijske suspenzije

Krivulja rasta ima četiri različite faze (slika 2)

1. Lag faza

Kada se mikroorganizam unese u svježi medij, potrebno je neko vrijeme da se prilagodi novom okruženju. Ova faza se naziva Lag faza, u kojoj se ubrzava ćelijski metabolizam, ćelije se povećavaju u veličini, ali bakterije se ne mogu replicirati i stoga nema povećanja ćelijske mase. Dužina faze kašnjenja direktno zavisi od prethodnog stanja rasta organizma. Kada se mikroorganizam koji raste u bogatom mediju inokulira u hranjivo siromašan medij, organizmu će trebati više vremena da se prilagodi novoj sredini. Organizam će početi sintetizirati proteine, koenzime i vitamine potrebne za njihov rast, pa će uslijediti povećanje faze zaostajanja. Slično, kada se organizam iz nutritivno siromašnog medijuma doda hranjivo bogatom medijumu, organizam se lako može prilagoditi okolini, može započeti diobu stanica bez ikakvog odgađanja, pa će stoga imati manju fazu kašnjenja, a može i izostati.

2. Eksponencijalna ili logaritamska (log) faza

U ovoj fazi mikroorganizmi su u stanju brzog rasta i dijeljenja. Njihova se metabolička aktivnost povećava i organizam započinje replikaciju DNK binarnom fisijom konstantnom brzinom. Medij za rast se eksploatira maksimalnom brzinom, kultura dostiže maksimalnu brzinu rasta i broj bakterija se povećava logaritamski (eksponencijalno) i na kraju se pojedinačne ćelije dijele na dvije, koje se repliciraju u četiri, osam, šesnaest, trideset dvije itd. (To je 2 0, 2 1, 2 2, 2 3. 2 n, n je broj generacija) To će rezultirati uravnoteženim rastom. Vrijeme potrebno bakterijama da se udvostruči u određenom vremenskom periodu poznato je kao vrijeme stvaranja. Vrijeme generiranja ima tendenciju da varira kod različitih organizama. E.coli dijeli se svakih 20 minuta, stoga je njegovo vrijeme generiranja 20 minuta, i za Staphylococcus aureus to je 30 minuta.

3. Stacionarna faza

Kako bakterijska populacija nastavlja rasti, mikroorganizmi troše sve hranjive tvari u mediju za rast za njihovo brzo razmnožavanje. To rezultira nakupljanjem otpadnih tvari, toksičnih metabolita i inhibitornih spojeva poput antibiotika u mediju. Time se mijenjaju uvjeti medija, poput pH i temperature, čime se stvara nepovoljno okruženje za rast bakterija. Brzina razmnožavanja će se usporiti, stanice podvrgnute diobi jednake su broju stanične smrti, i na kraju bakterija potpuno zaustavlja svoju diobu. Broj ćelija se ne povećava i time se brzina rasta stabilizira. Ako se ćelija uzeta iz stacionarne faze unese u svježi medij, ćelija se može lako kretati u eksponencijalnoj fazi i moći će obavljati svoje metaboličke aktivnosti kao i obično.

4. Faza opadanja ili smrti

Iscrpljivanje hranjivih tvari i kasnije nakupljanje metaboličkih otpadnih proizvoda i drugih otrovnih materijala u medijima olakšat će bakteriji prelazak u fazu smrti. Za to vrijeme bakterija potpuno gubi sposobnost razmnožavanja. Pojedine bakterije počinju umirati zbog nepovoljnih uvjeta, a smrt je brza i ujednačena. Broj mrtvih ćelija premašuje broj živih ćelija. Neki organizmi koji mogu odoljeti ovom stanju mogu preživjeti u okolišu stvaranjem endospora.

Slika 2: Različite faze rasta bakterije

IZRAČUN:


Vrijeme generiranja može se izračunati iz krivulje rasta (slika 3).

Slika 3: Izračunavanje vremena proizvodnje

Tačno udvostručene tačke iz očitavanja apsorbancije su uzete i tačke su ekstrapolirane da zadovolje odgovarajuću vremensku osu.


Vrijeme generiranja = (Vrijeme u minutama za postizanje apsorbancije 0,4) & ndash (Vrijeme u minutama za postizanje apsorbancije 0,2)

Neka je Ne = početni broj stanovništva


N = broj generacija u vremenu t



Stopa rasta može se izraziti kroz konstantu srednje stope rasta (k), broj generacija po jedinici vremena.



Srednje vrijeme generiranja ili srednje vrijeme udvostručenja (g) je vrijeme potrebno za udvostručavanje njegove veličine.

Zamjena jednadžbe 4 u jednadžbi 3


(Budući da se populacija udvostručuje t = g)

Konstanta srednje brzine rasta,
Srednje vrijeme proizvodnje,


Kako mogu usporediti krivulje rasta bakterija? - Biologija

Ostali dijelovi ove web stranice objašnjavaju kako se rade uobičajeni statistički testovi. Evo vodiča za odabir pravog testa za vaše potrebe. Kada ga pronađete, kliknite na & quotviše informacija?& quot kako biste potvrdili da je test prikladan. Ako znate da je prikladan, kliknite na & quotkreni!& quot

Bitan: Vaši podaci možda nisu u odgovarajućem obliku (npr. Postoci, proporcije) za test koji vam je potreban. To možete prevladati jednostavnom transformacijom. Ovo uvijek provjerite - kliknite ovdje.

Koristite ovaj test za usporedbu sredstava dva uzorka (ali pogledajte test 2 u nastavku), čak i ako imaju različit broj ponavljanja. Na primjer, možda biste htjeli uporediti rast (biomase, itd.) Dviju populacija bakterija ili biljaka, prinos usjeva sa ili bez tretmana gnojiva, optičku gustoću uzoraka uzetih iz svake od dvije vrste otopine itd. Ovaj test se koristi za "podatke o mjerenjima" koji su stalno promjenjivi (bez fiksnih granica), a ne za brojeve od 1, 2, 3 itd. Morali biste transformirati postotke i proporcije jer imaju fiksne granice (0-100 ili 0 -1).

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Koristite ovaj test poput t-test, ali u posebnim okolnostima - kada možete organizirati dva skupa repliciranih podataka u parove. Na primjer: (1) u pokusu usjeva, upotrijebite "quotplus & quot" i "quotminus" quot dušične kulture na jednoj farmi kao par, "quotplus" i "quotminus" quot dušične kulture na drugoj farmi kao par, i tako dalje (2) u ispitivanju droga gdje liječenje lijekovima se uspoređuje s placebom (bez liječenja), jedan par mogu biti 20-godišnji bijeli muškarci, drugi par mogu biti 30-godišnje azijske ženke itd.

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Koristite ovaj test ako želite uporediti nekoliko tretmana. Na primjer, rast jedne bakterije na različitim temperaturama, učinci nekoliko lijekova ili antibiotika, veličine nekoliko vrsta biljaka (ili zubi životinja itd.). Također možete usporediti dvije stvari istovremeno - na primjer, rast 3 bakterije na različitim temperaturama itd. Kao t-test, ovaj test se koristi za "podatke o mjerenjima" koji su stalno promjenjivi (bez fiksnih granica), a ne za brojeve 1, 2, 3 itd. Morali biste transformirati postotke i proporcije jer imaju fiksne granice (0-100, ili 0-1).

Više informacija? Ovo vam je potrebno jer postoje različiti oblici ovog testa.

Koristite ovaj test za usporedbu brojeva (brojeva) stvari koje spadaju u različite kategorije. Na primjer, broj ljudi plavookih i smeđeokih u klasi ili broj potomaka (AA, Aa, aa) iz eksperimenta genetskog ukrštanja. Također možete koristite test za kombinaciju faktora (npr. učestalost plavih/smeđih očiju kod ljudi sa svijetlom/tamnom kosom ili broj stabala hrasta i breze sa ili bez određene vrste žabokrečine ispod njih na različitim tipovima tla, itd.).

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Koristite ovaj test za postavljanje granica pouzdanosti na srednju vrijednost broja slučajnih događaja, tako da mogu se uporediti različita sredstva brojanja za statističku razliku. Na primjer, broj bakterija izbrojan u različitim kvadratima komore za brojanje (hemocitometar) trebao bi slijediti nasumičnu raspodjelu, osim ako se bakterije međusobno ne privlače (u tom slučaju bi brojevi na nekim kvadratima trebali biti abnormalno visoki, a na drugim kvadratima abnormalno niski ) ili se međusobno odbijaju (u tom slučaju brojanje bi trebalo biti nenormalno slično u svim kvadratima). Vrlo je malo stvari u prirodi nasumično raspoređeno, ali to bi pokazalo testiranje snimljenih podataka u skladu s očekivanjima Poissonove distribucije. Korištenjem Poissonove distribucije imate snažan test za analizu da li su objekti/ događaji nasumično raspoređeni u prostoru i vremenu (ili, obrnuto, jesu li objekti/ događaji grupisani).

Ovi postupci se koriste za sagledavanje odnosa između različitih faktora, i (ako je primjereno) za grafikovanje rezultata na statistički smislene načine. Na primjer, kako se temperatura (ili pH itd.) Povećava, raste li ili raste stopa rasta? S povećanjem doze lijeka raste li stopa odgovora pacijenata? S povećanjem nadmorske visine povećava li se ili smanjuje broj leptira (ili hrastova)? Ponekad odnos je linearan, ponekad logaritamski, ponekad sigmoidanitd. Možete testirati sve ove mogućnosti i u ispitivanjima lijekova ili toksičnosti (na primjer) izračunati LD50 ili ED50 (smrtonosna doza, ili procijenjena doza, za stopu odgovora od 50%).

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==========================================

Više informacija

Koristite ovaj test za usporedbu sredina dviju populacija koje ste uzorkovali (ali pogledajte test 2 u nastavku). Na primjer, možda ćete htjeti uporediti rast (biomase, itd.) Dviju bakterija ili biljaka, prinos usjeva sa ili bez dodanog dušika, optičku gustoću uzoraka uzetih iz svake od dvije vrste otopine itd.

Šta će vam trebati za ovaj test: minimalno 2 ili 3 ponavljanja svakog uzorka ili tretmana, ali idealno najmanje 5 ponavljanja. Na primjer, prinos izmjeren za 5 polja usjeva oplođenog dušikom i za 5 neoplođenih polja, optička gustoća 5 epruveta svake otopine, mjerenje 5 biljaka svake vrste itd. Velike veličine uzorka (10 ili više) uvijek bolji od malih uzoraka, ali lakše je izmjeriti visinu 10 ili 20 (ili 50) biljaka nego postaviti 10 ili 20 velikih fermentora!

Ne treba ti isti broj ponavljanja svakog tretmana - na primjer, možete uporediti 3 epruvete jedne otopine sa 4 epruvete druge. Ovaj test možete koristiti i za usporedbu nekoliko ponavljanja jednog tretmana s jednom vrijednošću za drugi tretman, ali on ne bi bio jako osjetljiv.

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Nazad na & quotKoji test mi treba? & Quot

Koristite ovaj test poput t-test, ali u posebnim okolnostima - kada možete organizirati dva skupa repliciranih podataka u parove. Na primjer: (1) u pokusu usjeva, upotrijebite "quotplus & quot" i "quotminus" quot dušične kulture na jednoj farmi kao par, "quotplus" i "quotminus" quot dušične kulture na drugoj farmi kao par, i tako dalje (2) u ispitivanju droga gdje liječenje lijekovima se uspoređuje s placebom (bez liječenja), jedan par mogu biti 20-godišnji muškarci, drugi par mogu biti 30-godišnje žene itd.

Zašto koristimo test uparenih uzoraka? Budući da su farme ili ljudi ili mnoge druge stvari inherentno varijabilne, ali uparivanjem tretmana možemo ukloniti veći dio ove slučajne varijabilnosti iz testa "kvotinitrogen u odnosu na bez dušika" ili "tretman lijekom u odnosu na bez tretmana" itd.

Koji su zahtevi za ovaj test? Glavni zahtjev je da se eksperiment PLANIRA unaprijed. Zatim možete upotrijebiti test uparenih uzoraka u mnoge svrhe - na primjer, dva tretmana u odnosu na jedan dan, zatim ista dva tretmana u usporedbi sljedećeg dana, itd.

Općenito, hoćete treba više replika nego za t-test (recimo, najmanje 5 za svaki tretman), i hoćete potreban je isti broj ponavljanja za svaki tretman.

Ali morate imati dobar razlog za uparivanje tretmana - ne biste to trebali činiti proizvoljno.

Samo napred!
Nazad na & quotKoji test mi treba? & Quot

Analiza varijanse za usporedbu sredstava tri ili više uzoraka.

Koristite ovaj test ako želite uporediti nekoliko tretmana. Na primjer, rast jedne bakterije na različitim temperaturama, učinci nekoliko lijekova ili antibiotika, veličine nekoliko biljaka (ili zubi životinja itd.). Također možete usporediti dvije stvari istovremeno - na primjer, rast 3 ili 4 soja bakterija na različitim temperaturama, itd.

Najjednostavniji oblik ovog testa je jednosmjerna ANOVA (ANALIZA VARIANCIJE). Koristite ovo za usporedbu nekoliko zasebnih tretmana (npr. efekti 3 ili više temperatura, nivoa antibiotika, tretmana usjeva itd.). Ti ces trebaju najmanje 2 ponavljanja svakog tretmana.

Jednosmjerna ANOVA vam govori postoje li razlike između tretmana kao cjelina. No, može se koristiti i oprezno, poput višestrukog t-testa, da vam kažem koji se tretmani međusobno razlikuju.
Ići na jednosmjernu ANOVU?
Nazad na & quotKoji test mi treba? & Quot

Drugi oblik ovog testa je dvosmjerna ANOVA. Koristite ovo ako želite za usporedbu kombinacija tretmana. Na primjer, za usporedbu rasta organizma na nekoliko različitih podloga na nekoliko različitih temperatura. Ili učinci dva (ili više) lijekova pojedinačno i u kombinaciji. Ili reakcije usjeva na tretiranje gnojiva na različitim farmama ili tipovima tla. Možeš dobiti korisne informacije čak i ako imate jednu od svake kombinacije tretmana, ali dobijate mnogo više informacija ako imate 2 (ili više) replika svake kombinacije tretmana. Tada vam test može reći imate li značajnu interaction - na primjer, ako se promjenom temperature promijeni način na koji organizam reagira na promjenu pH itd.
Ići na dvosmjernu ANOVU?
Nazad na & quotKoji test mi treba? & Quot

Upotrijebite ovaj test za usporedbu brojeva (brojeva) stvari koje spadaju u različite kategorije. Na primjer, za usporedbu broja plavookih i smeđeokih ljudi u klasi ili broja potomaka (AA, Aa, aa) iz eksperimenta genetskog ukrštanja. Također možete koristite test za sagledavanje kombinacija faktora (npr. učestalost plavih/smeđih očiju kod ljudi sa svijetlom/tamnom kosom ili broj žabokrečina ispod hrasta i breze na različitim tipovima tla itd.).

Za ovaj test upoređujete stvarne brojeve (u različitim kategorijama) sa & quotexpected & quot skupom brojanja. Ponekad su očekivanja očita - na primjer, da će polovina potomaka iz ukrštanja roditelja Aa i aa imati genotip Aa, a polovica će imati aa. Morate izgraditi hipotezu (nazvanu nulta hipoteza) koristeći logičke argumente.

Koji su zahtevi za ovaj test? Gotovo bilo koja vrsta podataka & quotcount & quot može se analizirati hi-kvadratom, ali morate koristiti & quotreal & quot brojeve, a ne proporcije ili postotke.

Samo napred!
Nazad na & quotKoji test mi treba? & Quot

Glavni zahtjev za ovaj test je da prosječan broj (kolonija bakterija, ljutičica itd.) Mora biti relativno visok (recimo 30 ili više) prije nego što se može očekivati ​​da će se prilagoditi Poissonovoj distribuciji. Ako imate tako veliki broj, tada možete provjeriti jesu li vaši rezultati u skladu s Poissonovom distribucijom.

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Nazad na & quotKoji test mi treba? & Quot

  1. Plot your results on graph paper, and ask yourself: does the relationship look (or is expected to be) linear, or is it logarithmic, or sigmoid (S-shaped)? You might need to transform the data (see transforming data) if they are not linear.
  2. Calculate the correlation coefficient, which tells you whether the data fit a straight line relationship (and how close the fit is, in statistical terms).
  3. If the correlation coefficient is significant, and other conditions are met, proceed to regression analysis, which gives the equation for the line of best fit, then draw this line on your graph.

1. Proportions and percentages: convert to arcsin values

Certain mathematical assumptions underly all the statistical tests on this site. The most important assumption is that the data are normally distributed and are free to vary widely about the mean - there are no imposed limits. Clearly this is not true of percentages, which cannot be less than 0 nor more than 100. If you have data that are close to these limits, then you need to transform the original data before you analyse them.

One simple way of doing this is to convert the percentages to arcsin values and then analyse these arcsin values. The arcsin transformation moves very low or very high values towards the centre, giving them more theoretical freedom to vary.

[You convert percentages (x) to arcsin values ( q ), gde q is an angle for which sin q is x/100 ]

On a calculator:

to get the arcsin value for a postotak (e.g. 50%), divide this by 100 ( = 0.5), take the square root (= 0.7071), then press "sin-1" to get the arcsin value (= 45). [NB: if your calculator gives the result as 0.785 then this is the angle in radians rather than degrees]

to get the arcsin value for a proporcija (e.g. 0.4), take the square root (= 0.6325), then press "sin-1" to get the arcsin value (= 39.23).

On an "Excel" spreadsheet:

convert percentages to arcsin values (and back again) by entering a formula into the spreadsheet - Go for it!

2. Logarithmic transformation

Use this for two purposes:

  • When fitting a curve to logarithmic data (exponential growth of cells, etc). Take the logarithm of each "growth" value and plot this against time (real values). You can use either natural logarithms or logs to base 10. The data should now show a straight-line relationship and can be analysed using correlation coefficient and regression.
  • In Analysis of Variance, when comparing means that differ widely. The reason for this is that an analysis of variance is based on the assumption that the variance is the same across all the data. But usually this will not be true if some means are very small and others are very large - the individual data points for the large mean could vary widely. [For example, a mean of 500 could be made up from 3 values of 100, 400 and 1000, whereas a mean of 50 could not possibly include such wide variation] This problem is overcome by converting the original data to logarithms, squeezing all the data points closer together. Contrary to expectations, this would show significant differences between small and large means that would ne be seen otherwise.

3. Converting Percentages to Probits

Some types of data show a sigmoid (S-shaped) relationship. A classic case is in dosage-response curves, for testing antibiotics, pharmaceuticals, etc. To analyse these relationships the "percentage of patients/cells responding to a treatment" can be converted to a "probit" value, and the dosage is converted to a logarithm. This procedure converts an S-shaped curve into a straight-line relationship, which can be analysed by correlation coefficient and regression analysis in the normal way. From the straight-line equation, we can calculate the LD50, ED50, i tako dalje.

The method for doing this in "Excel" is shown below.

The table below shows part of a page from an ‘Excel’ worksheet. Columns are headed A-F and rows are labelled 1-21, so each cell in the table can be identified (e.g. B2 or F11). Representative % values were inserted in cells B2-B21.

You will now see how to convert these % values into probits or arcsin values, and back again. If you do the relevant conversion in your own spreadsheet, you can then use the probit or arcsin values instead of % values for the statistical tests.

In cell C2 of the spreadsheet. a formula was entered to convert Percentage to Probit values.

The formula (without spaces) is: =NORMINV(B2/100,5,1)

This formula is not seen. As soon as we move out of cell C2 it automatically gives the probit value (in C2) for the percentage in cell B2, seen in the "printout" below. Copying and then pasting this formula into every other cell of column C produces a corresponding probit value (e.g. cell C3 contains the probit of the % in cell B3).

Next, a formula was entered in cell D2 to convert Probit to Percentage, and the above procedure was repeated for all cells in column D.


Rezultati

Model-based prediction of time derivatives of bacterial growth and fluorescence

To predict the time derivatives of the cell number of and the number of fluorescent GFP in a bacterial culture, we adopted two simple models [26, 27] described in the “Methods and Materials” for the bacterial growth and GFP expression, as illustrated in Fig 2A and 2B. The bacterial growth model (Eqs 1 and 2) with three parameters (the activation rate α, the maximum growth rate k0, and the maximum possible bacterial number in a culture N) successfully produced the classical sigmoid growth curve (Fig 2C, blue solid line, with α = 1 hr -1 , k0 = 1 hr -1 , and N = 2×10 9 ). In addition, the GFP expression model (Eqs 3 and 4) with four parameters (the generation rate g, the maturation rate km, the degradation rate γ, and the degradation capacity M) predicted that the fluorescence of a bacterial culture kept increasing without reaching plateaus (Fig 2C, green dashed line, with g = 100 hr -1 , km = 1.5 hr -1 , γ = 500 hr -1 , and M = 2×10 11 ), resembling well with the experimental measurements from a microplate reader (Fig 1A, green squares). Note that, while reasonable values from the literature [7, 24, 26, 27] were used for the parameters in this example, detailed studies by varying the model parameters were performed in the sections below.

From the simulated curves of n(t) i strf(t), we then examined their time derivatives numerically. The first-order time derivative of the cell number (dn/dt) showed a bell-shaped peak (Fig 2D, orange dotted line), while the first-order time derivative of the number of GFP f (dpf/dt) showed a sigmoid shape (Fig 2E, purple dotted line), similar to the curve of n(t) (Fig 2C, blue solid line) and suggesting the possibility of using the fluorescence to report the growth of the bacteria. In addition, the second-order time derivative (d 2 strf/dt 2 ) showed a bell-shaped peak (Fig 2E, red dash-dotted line) and appeared very similar to dn/dt. We note that there is a small horizontal shift between the two peaks (Fig 2F), which was further investigated in more details as shown below.

Dependence of time derivatives on the activation rate α

As the activation rate describes how quickly the dormant bacteria transit into the growing, dividing, active ones, it influences the time a bacterial culture takes to adjust to the environment, and thus determines the lag time of the bacterial growth [27]. Therefore, it is expected that a higher activation rate results in shorter lag time and thus a lower location of the peaks of the time derivatives (i.e., dn/dt i d 2 strf/dt 2). To quantitatively estimate the dependence of the time derivatives on the activation rate α, we solved the Eqs (1)–(4) and performed numerical simulations with varying α from 10 −4 to 10 2 hr -1 , while keeping the other parameters constant (k0 = 1 hr -1 , N = 2×10 9 , g = 100 hr -1 , km = 1.5 hr -1 , γ = 500 hr -1 , and M = 2×10 11 ). As expected, the cell-number curve shifted to the left as the activation rate increased (Fig 3A), while the fluorescence growth curve showed a similar trend (Fig 3B). The simulated results showed bell-shaped peaks in the time derivatives (Fig 3C and 3D). We found that increasing the activation rate led to horizontal shift of the peaks to the left while the peak heights did not change. We quantitatively determined the peak locations (i.e., the time points corresponding to the maxima of the peaks, τstr and , respectively) and observed that both τstr and exponentially decreased as the activation rate increased for α≤1 hr -1 (Fig 3E, black circles and green triangles), while the peak locations were much less sensitive to higher α (≥1). In addition, we observed that the difference between and τstr (i.e., ) remained constant with varying α (Fig 3E and S2A Fig in S1 File), indicating that the activation of the bacteria from dormancy to the active growing state did not play a role in the horizontal shift between the OD-based peak and fluorescence-based peak observed in Fig 2F.

Predictions of the models for the dependence of the cell number n the number of GFP f strf, and their time derivatives on the activation rate α. (A-D) Predicted curves of the models for the (A) n, (B) strf, (C) dn/dt, and (D) d 2 strf/dt 2 as functions of time with increasing activation rates α ranging from 10 −4 to 10 +2 hr -1 . (E) Predicted dependence of the peak locations (τstr–black circles and –green triangles) of the dn/dt i d 2 strf/dt 2 curves on the activation rate α, compared to that of the fitted lag time λ (blue squares). (F) Relation between the peak locations (τstr–black circles and –green triangles) and the fitted lag time λ. Blue dashed line and red dotted line are linear fittings.

To further confirm that the peak locations (τstr and ) were capable of reporting the lag time (λ) of the bacterial culture, we fitted the logarithm of the simulated n(t) curves by the Gompertz model, , where μ is the maximum specific growth rate and λ is the lag time [19, 24]. Comparing the fitted lag times (Fig 3E, blue squares) with the peak locations showed that the dependence of the lag time λ on the activation rate was the same as the peak locations, although the vertical baselines were different. We also observed that both the peak locations were linear to the lag time (Fig 3F) with the same slope that is close to one (1.12 ± 0.01), suggesting that the peak locations could be used to report the lag time of the bacterial growth, although a slight inflation exists (as the slope is 1.12 > 1).

Dependence of time derivatives on the maximum growth rate k0

The maximum growth rate k0 in the model describes how fast the bacteria divide in the presence of unlimited nutrients and resources [27] therefore, it is expected that k0 affects the heights of the peaks in the time derivatives. We ran simulations by varying k0 from 0.5 to 2.0 hr -1 , while keeping the other parameters constant (α = 1 hr -1 , N = 2×10 9 , g = 100 hr -1 , km = 1.5 hr -1 , γ = 500 hr -1 , and M = 2×10 11 ). Kao što je očekivano, n(t) became steeper as the growth rate k0 increased (Fig 4A), resulting in higher peaks in dn/dt (Fig 4C). We also observed that the peak locations shifted to the left at higher k0 (Fig 4C). Similar effects were observed for d 2 strf/dt 2 (Fig 4B and 4D). The peak height (ηstr) od dn/dt was linear to the growth rate k0 (Fig 4E, black circles), similar to the μk0 relation where μ is the maximum specific growth rate from the Gompertz fitting (as observed in the inset of Fig 4E). However, the dependence of the peak height ( ) of d 2 strf/dt 2 deviated significantly from a line (Fig 4E, green triangles). Interestingly, the power law, , (i.e., linear in a log-log scale, ) was more appropriate. Fitting the data gave an exponent of 0.84 (Fig 4E, red dotted line). It is noted that performing the power law fitting on ηstr resulted in an exponent of 1.00, confirming the linearity between ηstr i k0 (Fig 4E, blue solid line). We also observed that the relation between the peak locations (τstr and ) and the growth rate k0 followed the power law with exponents of -0.91 and -0.85, respectively (Fig 4F, black circles and green triangles). Note that the observed large changes of the peak locations (

7 hr from k0 = 0.5 to 2.0 hr -1 ) did not necessarily correspond to large changes in the lag time λ. In contrast, fitting with the Gompertz model [19, 24] showed a much smaller change (≤0.5 hr) in the lag time for the same k0 range (Fig 4F, blue squares). We also observed that the difference between and τstr showed slight dependence on the maximum growth rate k0 (S2B Fig in S1 File).

(A-D) Predicted curves of the models for the (A) n, (B) strf, (C) dn/dt, and (D) d 2 strf/dt 2 as functions of time with increasing growth rates k0 ranging from 0.50 to 2.00 hr -1 . (E) Predicted dependence of the peak heights (ηstr–black circles and –green triangles) of the dn/dt i d 2 strf/dt 2 curves on the growth rate k0 in the model. Inset: predicted dependence of the fitted maximum specific growth rate μ on the growth rate k0 in the model. (F) Predicted dependence of the peak locations (τstr–black circles and –green triangles) of the dn/dt i d 2 strf/dt 2 curves and the fitted lag time (λ –blue squares) on the growth rate k0. Blue dashed line and red dotted line are exponential fittings.

Dependence of time derivatives on the GFP expression rate g and maturation rate km

We then examined how the time derivatives depend on the GFP expression rate g and maturation rate km. Since these rates are not related to the activation and division of bacteria, the cell number does not depend on these two parameters (see Eqs 1 and 2). Therefore, dn/dt is independent on g i km. On the other hand, we expected that these two rates significantly affect the number of fluorescent GFP strf in the bacterial culture [26] and thus its time derivatives. To test this hypothesis, we ran simulations with either varying the expression rate or the maturation rate (one at a time) while keeping the other parameters constant. As shown in Fig 5A, strf became much steeper as the expression rate g increased, resulting in higher peaks in the second-order time derivative d 2 strf/dt 2 (Fig 5A, inset). However, the peak location was not affected (Fig 5A, inset) therefore, was independent on the expression rate g (Fig 5B and S2D Fig in S1 File). More quantitatively, we found the peak height was linearly dependent on the expression rate when all the other parameters were kept constant (Fig 5B). When varying the maturation rate km, we observed that not only the peak heights but also the peak locations were affected. As shown in Fig 5C, strf became steeper at higher maturation rates, resulting in higher peaks in the time derivatives (Fig 5C, inset). In addition, the peaks were shifted to the left as the maturation rate increased (Fig 5C, inset). Quantifying the peak location and height showed that the dependence of the peak location on the maturation rate km followed a power law, with an exponent of -0.11 (Fig 5D, blue triangles). As a result, the difference between and τstr was significantly dependent on the maturation rate km (Fig 5D and S2E Fig in S1 File), suggesting that the horizontal shift between the fluorescence-based peak and the OD-based peak observed in Fig 2F was largely due to the maturation of the fluorescent proteins. Interestingly, the relation between the peak height and the maturation rate km resembled the Michaelis–Menten kinetics [33] curve (Fig 5D, black circles), although the underlying reason is unclear. The simulated curve could be fitted well with , giving KM = 0.26 hr -1 .

(A) Predicted curves of the models for strf i d 2 strf/dt 2 (inset) as functions of time with increasing generation rate g ranging from 1 to 1.1×10 3 hr -1 . (B) Predicted dependence of the peak height ( –black circles) and peak location ( –blue triangles) of the d 2 strf/dt 2 curve on the GFP generation rate g. (C) Predicted curves of the models for strf i d 2 strf/dt 2 (inset) as functions of time with increasing maturation rate km ranging from 0.04 to 2.56 hr -1 . (D) Predicted dependence of the peak height ( –black circles) and peak location ( –blue triangles) of the d 2 strf/dt 2 curve on the GFP maturation rate km. Green solid line and red dashed line are fittings.

Dependence of time derivatives on the GFP degradation

We lastly investigated how the second-order time derivative of the fluorescence curve depends on the GFP degradation by running simulations with either varying degradation rate γ or varying degradation capacity M (one at a time) while keeping all the other parameters constant. We observed that the number of fluorescent GFP strf became shallower as the degradation rate γ increased, resulting in a decrease in the peak height ( ) of d 2 strf/dt 2 (Fig 6A). More interestingly, at high enough degradation rate, the strf(t) curve started to show slower growth at longer times or even plateaus (e.g., 4×10 11 hr -1 ) (Fig 6A). This change in the growth curve was reflected by the dip after the peak in d 2 strf/dt 2 and a slight left shift in the peak location (Fig 6A, inset). Quantifying the peak height and location showed that both of the peak height and peak location were almost constant below γ<10 10 hr -1 (Fig 6B inset, with M = 2×10 11 ) but decreased quickly at higher degradation rates (Fig 6B). For the degradation capacity (M), we observed higher M values in general gave steeper strf curves (Fig 6C), which is expected as a larger degradation capacity (but the same degradation rate γ) indicates that more GFP proteins are degraded per unit time [26]. Interestingly, a more careful examination showed that the dependencies of both peak height and peak location on the degradation capacity were more complicated than monotonic changes (Fig 6C, inset). Quantifying the peak locations and heights showed that interesting dependence on the degradation capacity (Fig 6D). We also note that both the degradation rate and capacity affected , the horizontal shift between the fluorescence-based peak and OD-based peak (Fig 2F).

(A) Predicted curves of the models for strf i d 2 strf/dt 2 (inset) as functions of time with increasing degradation rate γ ranging from 100 to 2×10 12 hr -1 . (B) Predicted dependence of the peak height ( –black circles) and peak location ( –blue triangles) of the d 2 strf/dt 2 curve on the GFP degradation rate γ. Inset: a close-up look of the same data in the range of γ∈[1, 10 10 ] hr -1 . (C) Predicted curves of the models for strf i d 2 strf/dt 2 (inset) as functions of time with increasing degradation capacity M ranging from 10 8 to 10 15 . (D) Predicted dependence of the peak height ( –black circles) and peak location ( –blue triangles) of the d 2 strf/dt 2 curve on the GFP degradation capacity M.

Application of the time derivative based method to study lag time elongation caused by Ag + -treatment

We have been investigating the antimicrobial activity and mechanism of silver (Ag) in various forms, such as ions and nanoparticles [7, 30, 34, 35]. For example, we performed growth-curve measurements on E. coli bacteria in the presence of Ag + ions following standard protocols (i.e., using cuvettes and photometry), and found that the major effect of Ag + ions was to elongate the lag time [7]. Here we repeated the growth experiments of GFP-expressing E. coli bacteria in the absence and presence of Ag + ions at 40 μM with a multimode microplate reader by measuring both the optical density at 600 nm (OD) and green fluorescence (FL). We observed that, due to significant multiple scattering, the OD curve from the microplate reader did not follow the sigmoid shape (Fig 7A), even if the overnight culture (i.e., 16 hr) was expected to be in the stationary phase. This observation was consistent with previous results reported in the literature [2]. On the other hand, the growth curves at low OD values (e.g., ≤ 0.8–1.0) confirmed that the presence of Ag + ions elongated the lag time while keeping the growth rate (i.e., the slope) roughly the same (Fig 7A), reproducing our previous results [7]. More quantitatively, we fitted the logarithm of the OD curves partially (using data with OD ≤ 1.0) by the Gompertz model [24] and found that the lag times were 1.38±0.02 hr and 6.6±0.9 hr, respectively, corresponding to a lag time elongation of 5.2±0.9 hr. The reported errors here were fitting errors, which were large in some cases (e.g., +Ag + ions) presumably due to the missing of the bacterial growth data above 1.0.

(A) OD growth curves of E. coli in the absence (-Ag + , blue circles) and presence (+Ag + , orange squares) of 40 μM Ag + ions, measured with a multimode 96-well microplate reader. (B) FL growth curves of the same E. coli samples (-Ag + : green crosses Ag + : red triangles). Error bars in panels A i B represent standard errors of the means (SEM). (C, D) Time derivatives of the corresponding growth curves in panels A i B. Vertical lines highlights the corresponding peaks. (E) Measured peak locations of ΔOD/Δt in the absence ( ) and presence ( ) of Ag + ions. Error bars stand for the standard deviations of the peaks. (F) Measured peak locations of Δ 2 FL/Δt 2 in the absence ( ) and presence ( ) of Ag + ions. Error bars stand for the standard deviations of the peaks. (G) Comparison between the changes of the peak locations (Δτstr and ) and the elongation of the fitted lag time (Δλ).

We calculated the first-order time derivatives of the OD curves (ΔOD/Δt), which showed distinct peaks (Fig 7C). In the absence of Ag + ions, the peak was centered at 4 hr with a standard deviation (STDEV) of 1.2 hr (Fig 7C, blue circles). In contrast, the peak for the Ag + -treated bacteria was centered at 9.5 hr with a STDEV of 1.4 hr (Fig 7C, orange squares). The shift in the peak location due to the Ag + -treatment was 5.5 ± 1.9 hr (Mean ± STDEV, Fig 7E), consistent with the lag time elongation estimated from the Gompertz fitting (Fig 7G).

We also measured the fluorescence curves of the same samples in the absence and presence of Ag + ions (Fig 7B). The shapes of the measured FL curves were similar to the predictions from the models (Figs 2–6), providing evidence to support the validity of the models used in this work. More importantly, we calculated the second-order time derivative (Δ 2 FL/Δt 2 ) and observed the model-predicted peaks centered at 5.3 hr and 10.8 hr, respectively (Fig 7D). The shift in the peak location was 5.5 ± 1.1 hr (Mean ± STDEV, Fig 7F), corroborating very well with the shift from the OD results and the lag time elongation from the Gompertz fitting (Fig 7G).

Applying the time derivative based method to study the growth E. coli bacteria in the presence of AgNPs

When attempting to measure the OD growth curves of bacteria treated with PVP-coated cubic silver nanoparticles (AgNPs) at various concentrations (0–80 μg/mL) using the microplate reader, we observed that the OD curves were significantly affected by the presence of the AgNPs (Fig 8A, inset). First, we observed a shift in the baseline of the OD measurements, presumably due to the contributions of the AgNPs to the absorption and scattering of light. More importantly, strange peaks emerged in the OD curves at short time, possibly due to the aggregation, dissolution, and other processes of the AgNPs in the growth media. These peaks in the OD curves made it challenging to directly apply the Gompertz model to quantitatively obtain the growth properties of the bacteria, without making various assumptions about the peaks.

(A) FL growth curves of E. coli bacteria in the presence of AgNPs at 0 (control, blue circles), 20 (orange squares), 40 (green cross), 60 (red triangle), and 80 (purple triangle) μg/mL. Inset: the corresponding OD growth curves for the same samples. Error bars represent standard errors of the means (SEM). (B) Time derivatives of the FL growth curves (Δ 2 FL/Δt 2 ) in panel A. Vertical lines highlight the peak locations. (C, D) Measured dependence of the (C) peak locations and (D) peak heights on the concentration of AgNPs.

On the other hand, we found that the fluorescence curves followed the predictions from the models in the presence of the AgNPs (Fig 8A). We applied the time derivative based method on the FL curves and observed that the peak centers in Δ 2 FL/Δt 2 were shifted to the right at higher concentrations of AgNPs, along with decreases in the peak heights (Fig 8B). It is noted that Hanning smoothing of curves was performed to reduce noises here. Characterizing the peaks showed that the peak location increased linearly, and the peak height decreased exponentially, as the concentration of AgNPs increased.


Bacterial Division

Bacteria and archaea reproduce asexually only, while eukartyotic microbes can engage in either sexual or asexual reproduction. Bacteria and archaea most commonly engage in a process known as binarna fisija, where a single cell splits into two equally sized cells. Other, less common processes can include multiple fission, budding, and the production of spores.

The process begins with cell elongation, which requires careful enlargement of the cell membrane and the cell wall, in addition to an increase in cell volume. The cell starts to replicate its DNA, in preparation for having two copies of its chromosome, one for each newly formed cell. The protein FtsZ is essential for the formation of a septum, which initially manifests as a ring in the middle of the elongated cell. After the nucleoids are segregated to each end of the elongated cell, septum formation is completed, dividing the elongated cell into two equally sized daughter cells. The entire process or cell cycle can take as little as 20 minutes for an active culture of E. coli bakterije.


The Growth of Bacteria Cells &ndash Explained! (With Figure) | Micro Biology

An individual bacteria cell grows in size, when environmental conditions are favourable for its growth. Each cell grows to approximately double its size (Figure 2.15).

In case of spherical bacteria, the diameter of the cell doubles, while in others, the cell elongates to double its original length.

Such growth is called ‘cellular growth’. After a bacteria cell attains almost double its size, it divides into two cells by a process called ‘binary fission’. Thus, reproduction of bacteria takes place through binary fission. The term binary implies that each mother bacteria cell divides (fission: division) to two (bi: two) daughter bacteria cells.

During division, the cell membrane and cell wall at the middle of the mother cell grow inward from opposite sides till they meet each other and from a partition wall called ‘septum’.

The septum divides the cell into two equal halves, which later on pinch off to form two new daughter mother cell, its DNA molecule replicates to two similar DNA molecules, so that each daughter cell receives one DNA molecule. Other cellular substances are also divided equally between the two daughter cells.

Growth of Bacteria:

In case of higher plants and animals, growth implies an increase in the size of an individual. Although each bacteria cell also grows by increase in its size, such cellular growth is difficult to be perceived ordinarily and is of little importance rather it is the number of cells produced at the end of a certain time interval, which can be perceived and has definite importance.

That is why ‘growth of bacteria’ is defined as an increase in the number of bacteria cells. ‘Growth rate’ of bacteria is defined as the increase in the number of bacteria cells per unit time. The time required for a given population of bacteria to double is called ‘generation time’ or ‘doubling time’. It varies among bacteria from few minutes to few hours.

Exponential or Logarithmic Growth:

As growth of bacteria takes place through binary fission, a single bacterium (1) grows as 1,2,4,8,16 and so on, which can also be expressed as 1 x 2 0 , 1 x 2 1 , 1 x 2 2 , 1 x 2 3 , 1 x 2 4 ,……………….. 1 x 2 n respectively. This type of growth, in which the number of cells doubles during each unit time (generation time), is called ‘exponential growth’ or ‘logarithmic growth’. Logarithmic growth is much faster than arithmetic growth (1, 2, 3, 4, 5, 6, 7….) or geometric growth (1, 2, 4, 8, 16, 32……).

Though, apparently it follows geometric growth, after few generations it grows as 1, 10, 100, 1000, 10000………. (10 0 , 10 1 , 10 2 , 10 3 , 10 4 ……..) Whose logarithmic values are 0, 1, 2, 3, 4……..respectively?

If the initial number of bacteria is N0 instead of 1, then after ‘n’ number of generations, the final number of bacteria (N) will be N0x 2 n .

Thus, the final number of bacteria can be obtained using the following equation:

N: Final number of bacteria,

N0: Initial number of bacteria and

The equation to find out the number of generations (n) is derived from the above equation as follows:

=> Log N = log (N0 x 2 n ) (taking log of both the sides)

=> Log N = log N0 + log 2 n (… log a x b = log a + log b)

=> Log N = log N0 + n log 2 (… log a x = x log a)

Growth Curve:

Growth of bacteria takes place in four phases as given below. A plot of log of bacteria number versus time gives a typical curve called ‘growth curve’ (Figure 2.16).

When an inoculum of bacteria is inoculated into a suitable fresh culture medium, normal logarithmic growth usually does not begin immediately rather it starts after a certain lapse of time. This lapse of time between the inoculation and beginning of the normal logarithmic growth of bacteria is called ‘lag phase’.

During this period, the bacteria acclimatise to the new environment of the fresh culture medium, which is not the same as the environment, from which it has been taken. In this phase, the bacterium grows very slowly through division by binary fission. Therefore, in the growth curve, the lag phase slopes upward only slightly.

A lag phase usually does not occur, if the inoculums is taken from an exponentially growing culture and is inoculated into a fresh culture medium similar to that, from which it has been taken and maintained under similar conditions of growth.

2. Log Phase (Exponential Phase):

During this period, the bacterium grows at the fastest rate in a logarithmic (exponential) manner. Maximum growth takes place during this phase. Generation time and growth rate remain almost constant. Therefore, in the growth curve, the log phase shows a steep rise from the end of the lag phase.

In the stationary phase, the number of cells in the culture remains almost constant. An indefinite exponential growth is impossible and can be compared with the story of a poor beggar making fool of a king by asking for a simple alms double the match sticks every day for a year starting with one. (1,2,4,8,16,32,64,128,256,512,1024, 2048,4096,8192,16384, 32768, 65536,131072,262144,5 24288, 1048576, 2097152, 4194304, 8388608,16777216, 3354432, 6708864, 13417728, 26835456, 53670912,………………….. only in one month).

It has also been calculated that a bacterium weighing only 10 -12 gram and having a generation time of 20 minutes, if grows exponentially for 48 hours, would produce a population weighing about 4000 time the weight of the earth.

Exponential growth does not continue indefinitely and ceases after some time because of two reasons: a) The culture medium becomes so over-populated that, essential nutrients present in it are used up and become unavailable after sometime and b) Due to over-population, toxic waste metabolites produced by the bacteria accumulate to inhibitory levels.

These lead to the beginning of death of the bacteria cells in the culture. Although cells reproduce by binary fission and growth continues unabated, the number of cells produced almost equals the number of cells dying. This leads to the stationary phase.

4. Decline Phase (Death Phase):

In this phase, the number of bacteria cells in the culture decreases. As more and more toxic metabolites accumulate in the medium, more and more cells start dying. This leads to more cells dying than produced. As a result, the number of cells decreases. The death phase also occurs exponentially (logarithmically), but at a much slower rate than that of the exponential growth phase.


Differences encountered in a Real Laboratory:

In an actual laboratory setting, there are certain important steps that are not necessarily applicable in a virtual lab:

  • Always wear lab coat and gloves when you are in the lab. When you enter the lab, switch on the exhaust fan and Laminar Air Flow.
  • Make sure media required for the experiment are available. If it is not available, prepare the media and autoclave it.
  • Always label the plates and conical flask with:

1. The name of the organism
2. The type of media
3. Your initials
4. The date

  • Properly adjust the flame of the Bunsen burner. The proper flame is a small blue cone it is not a large plume, nor is it orange.
  • While flaming the loop (or needle), be sure that each segment of metal glows orange/red-hot before you move the next segment into the flame. Be careful the metal will get extremely hot.
  • Once you have flamed your loop (or needle), do not lay it down, blow on it, touch it with your fingers, or touch it to any surface other than your inoculums or the sterile media. If you do touch the tip to another surface or blow on it, you will have to re-flame the loop before you proceed with your inoculation.
  • Allow your loop or needle to cool before you try to pick up your organism. If you pick up organism with a hot tool, your cells will be killed. To cool your loop or needle quickly, place it on a section of agar that is uninoculated or is at least different from the area from which you will remove cells.
  • Ensure that you are transferring the correct organism into your labeled conical flask by double-checking the name of the organism on the stock culture from which you are collecting your inoculums.
  • When removing the caps (cotton plug) from conical flask, always keep the caps (cotton plug) in your hand. Never set them on the table, as they could pick up contaminants.
  • Always handle open flasks at an angle never let them point directly up, since airborne or other environmental organisms could fall into the tube and cause contamination. Also, keep the lid over a plate when removing inoculums, as this will help prevent environmental contamination.
  • As soon as you done inoculating, flame your loop or needle. Never place a contaminated tool on your workbench.
  • Always flame the lip of the conical flask when you open it and before you replace the cap.
  • As soon as you are done inoculating, flame your loop or needle. Never place a contaminated tool on your workbench.
  • While operating the colorimeter, make sure that water droplets do not enter the slot of the colorimeter as it will damage the instrument. So wipe the cuvette using a lint free tissue paper before inserting it into the slot.
  • Do not touch the part of the cuvette that gets inserted into the colorimetric slot with hands and ensure that the cuvette is free from water droplets.
  • Discard all contaminated materials properly and return your supplies to the proper storage locations, and clean up your working area.
  • Always disinfect your work area when you are finished.

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Stacionarna faza

The number of new cells produced balances the number of cells that die, resulting in a steady-state. In batch culture, exponential growth cannot occur indefinitely because the essential nutrients of the culture medium are used and waste products of organisms accumulate in the environment. In the stationary phase, there is no net increase or decrease in cell number. The cells function such as energy metabolism and some biosynthetic processes go on. (Bilješka: cells grown in chemostat do not enter the stationary phase)


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