Fitting The Negative Binomial Distribution Biology Essay

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Pasteuria penetrans is an obligate parasite of root-knot nematodes (Meloidogyne spp.). The bacterium is cosmopolitan, frequently encountered in many climates and environmental conditions, making it a promising candidate for the biological control of root-knot and other economically important plant parasitic nematodes (Sayre and Starr 1988; Hewlett and Dickson 1994).

The antagonistic potential of P. penetrans is reported to be influenced by several biotic and abiotic factors (Hatz and Dickson 1992; Davies et al., 1991; Verdejo-Lucas 1992; Ouri 1997). Soil variables like texture, moisture and temperature were shown by Talavera and Mizukubo (2003) as factors that favour nematode mobility in soil and so increase the chances of P. penetrans spore attachment to root-knot nematode juveniles as they search for host roots. Also isolates of P. penetrans have been recognised which adhere only to a particular species of root-knot nematode or to individual populations within a species (Stirling 1985; Davies and Danks 1992; Channer and Gowen 1992). These results and those of many others imply high variances in the numbers of spores attaching but no attempt has been made to examine this variability in detail.

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Overall, in this paper methods that model counts of P. penetrans spore attachment to Meloidogyne spp., are presented that examine in detail the variability of P. penetrans spore attachment and offer an explanation for it. Further we used a Markov chain model (Hoppensteadt, 1982; Berresford and Rockett, 2005; Waner and Costenoble, 2011), to describe nematodes movement process with or without P. penetrans spores.

Materials and methods

Meloidogyne spp. culture

A culture of Meloidogyne (M. javanica) was maintained on tomato plants (c.v. Tiny Tim) in the glasshouse. Eggs were collected and second stage juveniles (J2) were hatched using standard laboratory practices (Southey, 1986).

Pasteuria penetrans spore attachment bioassays

Spore suspensions of a commercial product of Pasteuria penetrans (Nematech Co. Ltd Japan) were prepared in tap water (Vagelas et al., 2011). Attachment tests on freshly hatched J2 were conducted in 2.5 cm Petri dishes using standard techniques (Davies et al., 1991). Data were recorded 1, 3, 6 and 9h after placing nematodes in the spore suspensions (Channer and Gowen, 1992).

For attachment bioassay fresh J2s of root knot nematodes were exposed to 5000 spores per Petri dish (Darban et al., 2004). All dishes were placed in a 28 0C incubator. Nematodes were observed under an inverted microscope at X 200 magnification and numbers of P. penetrans spores attached per nematode were recorded in three data sets. For the first (A), second (B) and third (C) data set (Table 1, Figure 1) a total of 36 nematodes were examined for P. penetrans spore attachment from each treatment, after incubation of the Petri dishes at 28 0C for 1h, 3h, 6h and 9h. For the fourth (D) and the fifth (E) data sets (Table 2 and 3) a total of 100 nematodes were examined for P. penetrans spore attachment from each treatment, after incubation of the Petri dishes at 28 0C for 1h, 3h, 6h and 9h.

Fitting the Negative Binomial Distribution to Pasteuria penetrans attachment

All calculations and graphs (based on observed data sets), for Poisson and Negative Binomial (NB) function were made on Excel spreadsheest.

Using the computer program BestFit 3.0 for Windows we a) estimate the best fit distribution (NB or Poisson) and b) measure how well the sample data (observed values) would fit a hypothesized probability density function of variables (theoretical values) using the chi-square test for goodness-of-fit.

Modelling probability of spores' attachment to root-knot juveniles with Markov chain.

A Markov decision process (Markov chain) was used to model the random variable (P. penetrans spores attachment) changes thought time (1, 3, 6 and 9h). This is based on the formula v(t)=v(t-1)A, (where A transition matrix and v(0)= initial probability vector), we computed the future probability distribution vectors for time t (t=1, 3, 6 and 9h). In this form v(t)=v(t-1)A, the ijth element of A is the conditional probability, Aij = P(System will be in state j at time t | It is in state i at time t-1) and each row of A, the elements sum to 1 (http://math.plussed.net/markov/). The dataset D was entered to an Excel spreadsheet to determine the transition probabilities and each transition matrix were calculated using standard matrix multiplication.

Using the Markov chain model to predict Pasteuria penetrans attachment

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From the above transition matrix we used the steady state distribution P12 to predict the attachment process for the fifth set (dataset E) of the observed data.

Results

Fitting the Negative Binomial Distribution to Pasteuria penetrans attachment

In studying P. penetrans spores attachment, a juvenile of root knot nematode (J2), may be encumbered with one or more spores over a fixed period of time. If every J2 were exposed equally to the chance of being encumbered with P. penetrans spores over a fixed period of time, the distribution would follow the Poisson series and the expected variance (s2) is equal to a mean. As figure 1 shows, the observed variance (s2) is significant larger that the mean on the basis of 3, 6 and 9h incubation periods. Therefore the successful events of P. penetrans attachment cannot be formulated as a Poisson process as the parameter s2 is not small or equal to the mean (Figure 1).

As shown in Figure 1, the means, e.g. at 9h of nematode exposure to a P. penetrans spore suspension, are smaller (twice less) than the variance indicating a strong overdispersion. This suggests that P. penetrans spores are clumped and more than one spore sticks on each J2. Further, the literature indicates that the 'over-dispersion' phenomenon is common for living organisms and can be explained by the Negative Binomial distribution (Ross and Preece, 1985).

Based on this, the data show that a better fit is obtained with the Negative Binomial distribution for P. penetrans spore attachment per juvenile at 3, 6 and 9h after application. In all those cases (3, 6 and 9h) we observed that the Negative Binomial distribution proved to be a better model to predict the observed (input) values (Table 1). The chi-square test of the hypothesis, shows that the Negative Binomial (NegBin) model was the most appropriate to fit the observed counts (Table 1).

Moreover, as the results of Figure 2 show, that the Negative Binomial is the more appropriate distribution fitting mostly all observations. Explanations for the Negative Binomial describing better the P. penetrans attachment are: a) the observed variance (s2) being larger than the mean (Figure 1) and b) as time increases the over-dispersion was clearly too large for the Poisson distribution.

Modelling probability of spores attaching to root-knot juveniles with Markov chain.

Table 2 shows the P. penetrans spore attachments per nematodes and nematodes counts encumbered with spores from dataset D at different times (1h, 3h, 6h and 9h).

Those data (Table 2), show that the best fit is obtained with the negative binomial distribution for P. penetrans spore attachment per juvenile at 3, 6 and 9h after application and the Poisson distribution is assumed for modelling for 1h exposure (Figure 3).

Further, based on the observed data of Table 2 encumbered J2s were summarized into four groups, J2s recorded with no P. penetrans spores (0 Pp), J2s encumbered with 1 to 2 P. penetrans spores (1 to 2 Pp), J2s encumbered with 3 to 7 P. penetrans spores (3 to 7 Pp) and J2s encumbered with ≥8 P. penetrans spores (≥8 Pp). Those group counts can be presented by the following transition matrix (1):

State

0 Pp

1 to 2 Pp

3 to 7 Pp

(≥8 Pp

1h

0.14

0.61

0.25

0

transition matrix (1)

3h

0.11

0.25

0.56

0.08

P=

6h

0.14

0.08

0.50

0.28

9h

0.08

0.09

0.36

0.47

The matrix 1 represents the probabilities of J2's encumbered with P. penetrans spores, showing the probabilities of transitions for encumbered with no spores, after 1h at 0.14 followed by, a probability for encumbered with 1 to 2 spores, after 1h at0.61, a probability for encumbered with 3 to 7 spores, after 1h at 0.25 and probability for encumbered with ≥8 spores, after 1h as zero. After 9 h the corresponding probabilities are 0.08, 0.09, 0.30 and 0.47 respectively.

Solving transition matrix (1), gives the steady state distribution:

State

0 Pp

1 to 2 Pp

3 to 7 Pp

≥8 Pp

1h

0.12

0.26

0.50

0.12

state distribution for n=3

3h

0.13

0.18

0.48

0.21

P2=

6h

0.12

0.17

0.43

0.28

9h

0.11

0.14

0.42

0.33

Solving transition matrix (1), gives the steady state distribution P12:

[q1 q2 q3 q4] = [0.1190 0.1756 0.4442 0.2612]

Based on the results obtained from the steady state distribution P12 (q1 q2 q3 q4) we note that the last results q4 (0.2612) is less than the 0.47 in the original matrix (transition matrix 1) for >8 spores after 9h suggesting that spores detached e.g. after 9h. This observation is in agreement with estimated number of observation of 9h (Figure 3) where the difference between the 6h and 9h are very small (Figure 3). To the best of our knowledge observation on the detachment of spores of P. penetrans from pre-parasitic second-stage juveniles of Meloidogyne spp., was reported by Ratnasoma et al., (1991), Giannakou and Gowen (2004), a phenomenon probably occurred due to the forces act on P. penetrans spores attachement (Ciancio, 1995).

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In conclusion, in the long term, (e.g. after 9h of incubation) 11.9% of J2s are without P. penetrans spores, 17,56% of J2s are with 1 to 2 P. penetrans spores, 44.42% of J2s are encumbered with 3 to 7 P. penetrans spores and 26.12% of J2s are encumbered with ≥8 P. penetrans spores.

Moreover, based on results obtained from the steady state distribution P12 the rate of attachment significantly changes with time and more J2s were encumbered with clumps of spores confirmed by the Negative Binomial distribution model presented in Figure 2.

Fitting the Markov chain model to Pasteuria penetrans attachment inputs

Table 3 shows the P. penetrans spore attachments per nematodes and nematodes counts encumbered with spores from dataset E at different treatments (1h, 3h, 6h and 9h).

Based on results obtained from Table 3 and data from column 9h (Table 3), we can calculate the probabilities of J2s encumbered with P. penetrans spores onto four mentions above groups as,

0 Pp

1 to 2 Pp

3 to 7 Pp

≥8 Pp

Q =

9h

0.11

0.11

0.49

0.29

Further, based on results obtained from the steady state distribution P12, the observed data of Q was fitted (Figure 4). Based on that, Figure 4 shows that the probability distribution vector P12 calculated from transition matrix (1) are good estimators of predicted the P. penetrans attachment process received from another data set (Table 3).

Discussion and Conclusions

This paper is concerned with fitting Poisson and Negative Binomial models to Pasteuria penetrans attachment. The application of the Poisson and the Negative Binomial modeling approaches for modeling biological data (plant and natural organisms) were first presented by Bliss (1953). In Bliss's paper evidence is provided to show that the biological models are characterized by a significantly larger variance than the mean. This is called "over dispersion". It was concluded that in analyzing biological counts for which the variance is significantly larger than the mean, the model of the Negative Binomial distribution is the most appropriate.

Moreover, in this paper, data is presented of the observed values of P. penetrans spore attachment to nematodes and the predicted values using the Poisson and the Negative Binomial distribution. In the bioassay, the procedure discussed above confirmed that the Poisson distribution is a satisfactory model to P. penetrans spore attachment on root-knot nematodes but only for the 1h or 3h exposure. Interestingly the mean is equal to the variance suggesting 'under-dispersion' the Poisson distribution is considered the most appropriate model to fit the data sets. Similar results on other organisms were presented by Bliss (1953) and Ross and Preece (1985). After 3h of exposure the negative binomial model is the more appropriate model to fit the data 'over-dispersion' (Bliss 1953; Cox 1983; Ross and Preece, 1985; Morel and Nagaraj, 1993; Gschobl and Czado, 2008; Shane, 2008). The negative binomial model is also the preferred model as time of exposure increased (e.g. 6 or 9h).

Examination of the variability using the negative binomial was an appropriate model to describe these two counts (mean/variance) especially as the variance is greater than the mean 'over-dispersion' (Bliss, 1953; Ross and Preece, 1985). 'Over-dispersion' arises when organisms are 'clumped', 'clustered' or 'aggregated' in space or time (Ross and Preece, 1985; Binet, 1986). Same data we observed in this research with P. penetrans spores that are distributed in clumps and probably clustered around encumbered nematodes.

After 6 or 9h high numbers of P. penetrans spores per nematode were observed and the Negative Binomial model provided a more efficient means of describing this. Possibly this is evidence of uneven distribution of spores in the suspension and some nematodes may encounter clumps of spores or clustered around encumbered nematodes.

Many models have been used to estimate the over-dispersion exhibited among organisms (Barbour and Kafetzaki, 1991). In this research it was noted that the negative binomial distribution is the most appropriate model to describe P. penetrans 'over-dispersion' as an aggregating organism, leading to the hypothesis that the bacteria are clumped and clustered under natural conditions the distribution of bacteria will be uneven and will often occur in aggregations.

Further Markov chain proved a good tool for predicting the P. penetrans attachment process even when J2s are encumbered with clumps of P. penetrans spores. It is believed that with the Markov decision process it is possible to estimate P. penetrans attachment related to time if the attachment process depends only on the distribution of the previous stage. Generally, as our data show the Markov model proves an easy computation method to predict the observation function of the counts. That point is useful to estimate P. penetrans process based on other parameters such as soil properties. Several authors have proposed the idea of Markov chains (Leslie, 1945; Leslie, 1948; Bailey, 1975; Nisbet and Gurney, 1982; Taylor and Karlin, 1998, Daley and Gani, 1999; Kot, 2001), as an efficient statistical test for many applications in biological modeling where future outcomes will predict from observed counts. Finally we can concluded that probably a Markov chain algorithm needs to be constructed to software programs as Matlab or Mathematica to produce output values based on the P. penetrans spores attachment process.