# Open writing projects/Sage and cython a brief introduction

## Contents

### Sage and Cython: a brief introduction

## Abstract

This is a quick introduction to Sage, a powerful new computational platform that builds on the strengths of Python. This article was directly inspired by Julius B. Lucks' "Python: All A Scientist Needs"; I recommend reading it first as it explains some of the attractions of Python and biopython.

Sage is a free and open-source project for computation of all sorts that uses Python as its primary language and "glue". One of the goals of Sage is to provide a viable free and open-source alternative to Matlab, Maple, and Mathematica. Sage unifies a great deal of open-source mathematical and statistical software; it includes biopython as an optional package and the statistics language R by default.

## Sage notebook interface

A key feature of Sage is its notebook web-browser interface.

Jose Unpingco has made a good short introductory video on the notebook interface that may help get a sense of what its like.

The notebook starts by default in native Sage mode, which is almost identical to python. However, other environments are available. The screenshot below shows a Sage notebook in R mode, with some simple R commands executed.

The @interact command in the Sage notebook provides an easy way to make simple GUIs to explore data. In the example below, a user can enter the URL of a fasta-formatted protein file and a PROSITE-style regular expression. Using biopython and the "re" module of python we can search for and display matches to the pattern. For this screenshot, I used proteins from the malaria-causing Plasmodium falciparum and a fragment of the transthyretin pattern (Prosite PS00768). This is just for illustrative purposes - I do not claim any significance for these matches.

<syntax type="python"> def PStoRE(PrositePattern):

""" Converts a PROSITE regular expression to a python r.e. """ rePattern = PrositePattern rePattern = rePattern.replace('-',)rePattern = rePattern.replace(' ',)rePattern = rePattern.replace('x','.') rePattern = rePattern.replace('{','[^') rePattern = rePattern.replace('}',']') rePattern = rePattern.replace('(','{') rePattern = rePattern.replace(')','}') return rePattern

from Bio import Fasta import re import urllib2 as U @interact def re_scan(fasta_file_url = 'http://www.d.umn.edu/~mhampton/PlasProtsRef.fa', pat = input_box('G - x - P - [AG] - x(2) - [LIVM] - x - [IV] ', type = str, width = 60)):

re_pat = re.compile(PStoRE(pat)) parser = Fasta.RecordParser() prot_file = U.urlopen(fasta_file_url) fasta_iterator = Fasta.Iterator(prot_file, parser = parser) for record in fasta_iterator: matches = re_pat.findall(record.sequence) if len(matches) != 0: html(record.title) html(matches) print

Here is another example of the interact command, along with some of Sage's 2D plotting. We take a human mitochondrial DNA sequence and plot the fraction of CG dinucleotides in a window of variable size. This fraction is divided by 16, to normalize it against the expected value in the case of independently, uniformly distributed nucleotides (obviously a poor model in this case).

<syntax type = "python"> from Bio import SeqIO import urllib2 as U import scipy.stats as Stat f = U.urlopen('http://www.d.umn.edu/~mhampton/HomoSmito.fa') p = SeqIO.parse(f, 'fasta') hsmito = p.next().seq.tostring() f.close() display_f = RealField(16) @interact def cgplot(window_length = slider([2^i for i in range(4,12)],default = 2^9)):

avg = [16.0*hsmito[x-window_length: x+window_length].count('CG')/(2*window_length-1) for x in range(window_length, len(hsmito) - window_length)] mean = display_f(Stat.mean(avg)) std = display_f(Stat.std(avg)) html('Ratio of CG dinucleotides in a window of size ' + str(2*window_length) + ' to the expected fraction 1/16

in the human mitochondrion.

Mean value: ' + str(mean) + '; standard deviation: ' + str(std)) show(list_plot(avg, plotjoined=True), ymin = 0, ymax = max(avg))

For more (mostly mathematical) examples of the @interact command, see the corresponding Sage interact wiki page.

## Cython

Sage initially used an alternative to SWIG (described in Julius's article) called Pyrex to compile Python code to C when performance concerns demanded it. Because they needed to extend Pyrex in various ways, they created a friendly fork of Pyrex called "Cython". I believe it is fair to say that Cython is the easiest way to create C code in Python.

Lets look at a toy example of Cython usage. Consider the following python function, which we can imagine is the inner loop of some more complex operation: <syntax type="python"> def slow_logistic(n,a,lamb = 3.82):

""" A slow logistic map. """ q = a for i in range(n): q = lamb*q*(1-q) return q

time a = slow_logistic(10^6,.5)

Time: CPU 6.17 s, Wall: 6.17 s </syntax> So the pure python version takes about 6 seconds for 10^6 iterations. If we tell Sage we'd like to use Cython, and declare variables to C types, we get quite a speedup:

<syntax type="python">
%cython
def cython_logistic(int n, float a, float lamb = 3.82):

cdef float q q = a for i in range(n): q = lamb*q*(1-q) return q

time a = cython_logistic(10^6,.5)

CPU time: 0.04 s, Wall time: 0.04 s </syntax>

So "Cythonizing" our loop sped it up by a factor of over 100. Often by converting a few crucial functions to Cython we get all the speed we need.

For a slightly more real-world example, suppose we want to interactively explore the behavior of a system of ODEs as we change parameters. The example below uses the "repressilator" system of Elowitz and Leibler, and a crude Euler's method (just for illustrative purposes!) to solve it. The behavior is not too bad for smaller runs but it gets a bit sluggish as we increase the simulation time.

<syntax type="python"> def m_der(m,p,a,a0,n):

return -m + a/(1+p^n) - a0

def p_der(m,p,b):

return -b*(p-m)

def der_vec(m_list,p_list,a,a0,n,b):

d_vec = [] for i in (0,1,2): d_vec.append(m_der(m_list[i],p_list[(i+1)%3],a,a0,n)) for i in (0,1,2): d_vec.append(p_der(m_list[i],p_list[i],b)) return d_vec

def repressilator_run(m_init = [.1,.2,.3], p_init = [.1,.2,.3], a_val = 6, b_val = .1, n_val = 2.1, t_end = 100):

a0_val = .01 t_0 = 0 stepsize = .1 t = 0 m_data = m_init,p_init,t while t < t_end: der = der_vec(m_data[-1][0],m_data[-1][1],a_val,a0_val,n_val,b_val) new_m = [m_data[-1][0][i] + stepsize*der[i] for i in (0,1,2)] new_p = [m_data[-1][1][i-3] + stepsize*der[i] for i in (3,4,5)] m_data.append([new_m,new_p,t]) t = t + stepsize return m_data

bdigits = RealField(16)

html('### Repressilator simulator

')@interact def rep_run(alpha = slider(0,20,.01,10), beta = slider([10^i for i in srange(-3,1,.1)], default = .1), exp_n = slider(1,10,.1,2.1), t_end = slider([10^i for i in srange(0,3,.1)],default = 2)):

test = repressilator_run(a_val = alpha, n_val = exp_n) html('beta = ' + str(bdigits(beta)) + '; alpha = ' + str(bdigits(alpha))+ '; exponent = ' + str(bdigits(exp_n))) print jsmath('\\frac{d p_{i}}{dt} = -\\beta (p_i - m_i), \ \\frac{d m_{i}}{dt} = -m_i + \\frac{\\alpha}{1 + p_i^{n}} + \\alpha_0') show(sum([list_plot([[x[2],x[1][i]] for x in test], hue = i/3.0, plotjoined=True) for i in (0,1,2)]), ymin = 0)

If we time a typical short run, it takes about .3 seconds. The moderately Cythonized form below takes about .02 seconds (all these timings are on the same machine).

<syntax type = "python"> %cython cdef m_der(m,p,a,a0,n):

return float(-m + a/(1+p**n) - a0)

cdef p_der(m,p,b):

return float(-b*(p-m))

cdef der_vec(m_list,p_list,a,a0,n,b):

d_vec = [] for i in (0,1,2): d_vec.append(m_der(m_list[i],p_list[(i+1)%3],a,a0,n)) for i in (0,1,2): d_vec.append(p_der(m_list[i],p_list[i],b)) return d_vec

def repressilator_run(m_init = [.1,.2,.3], p_init = [.1,.2,.3], a_val = 6, b_val = .1, n_val = 2.1, t_end = 100):

cdef float a0_val, t_0, stepsize, t a0_val = .01 t_0 = 0 stepsize = .1 t = 0 m_data = m_init,p_init,t while t < t_end: der = der_vec(m_data[-1][0],m_data[-1][1],a_val,a0_val,n_val,b_val) new_m = [m_data[-1][0][i] + stepsize*der[i] for i in (0,1,2)] new_p = [m_data[-1][1][i-3] + stepsize*der[i] for i in (3,4,5)] m_data.append([new_m,new_p,t]) t = t + stepsize return m_data

</syntax>

Note that the above code is still using quite a few python objects and methods, and could be sped up further with more effort.

There is much more to Sage but hopefully this introduction has whet your appetite.

## References

"A Synthetic Oscillatory Network of Transcriptional Regulators"; Michael B. Elowitz and Stanislas Leibler; Nature. 2000 Jan 20;403(6767):335-8.