Create an ORF generating HMM $(50$ points$)$

Create a THREE$-$STATE HMM that can simulate a sequence which contains Open Reading Frames$($ORFs$).$ It should run for a fixed number of emissions, x$,$ during every run of the model. This model should be able to generate ANY possible reading frame with non$-$zero probability$*.$ If an ORF ends before reaching x emissions your algorithm should continue the sequence by starting a new ORF. Your model should include the following parameters for which you need to make up reasonable values:

k$\_0$: a $3$ by $1$ matrix defining the probability of starting in each of your $3$ states,

A: A $3$x$3$ matrix containing the transition probabilities a$\_$kl which is the probility of transitioning from state k to state l

E$\_$k: one matrix per state representing probabilities for all possible emissions from this state.

x $=$ the total number of emissions that you would like your model to emit per run.

Hint: What are $3$ things that every ORF has?

$1)$ Draw the HMM $($by hand is fine$)$ and justify your parameter values $(25$ points$)$

a$.$ Model is drawn with E$\_$k$,$ and A values shown on the model. Within the bounds of your parameter $$x$,$ all possible ORFs can be generated by your model. $(15$ points$)$

b$.$ Reasonable probability values are chosen and an explanation accompanies the choice. $(10$ points$)$

$2)$ Code the model in Python$**.$ Use it to generate $250$ random sequences containing ORFs$***.$ Measure the lengths of the $250$ ORFs you generated and draw a distribution representing your result. Submit your code as $$yourName$\_$ORFHMM.py$$ submit the output of your code in $$your$\_$nameORFoutput.txt$$ and include the distribution in your pdf$.$

i$.$ Code correctly represents your HMM from question $1(10$ points$)$

ii$.$ ORF distribution is submitted $(5$ points$)$

iii. Is your distribution what you expected based on the model parameters you$$ve chosen? Explain. $(10$ points$)$

$*$ Use ATG as the only possible start. The probability may be small for some ORFs, but all should be non$-0$ for any ORF length of $($x$-1)$ where x is the total emissions of your model.

$**$ It$$s ok if your model generates a sequence that concludes with an incomplete ORF

$***$ For measuring the ORF lengths, use only the $*$FIRST$*$ complete ORF in each sequence even if each sequence has more than one complete ORF.