What is the relationship between dark current and the contribution of dark noise in a photodiode?
The Hamamatsu Photodiode Handbook  section 1.6 gives the equation as 

$$darknoise = \sqrt{2*q*Id*B}$$

where q is the charge of an electron, Id is the mean dark current, and B is the detection bandwidth.  

More intuitively, mean current divided by bandwidth is the expected number of electrons per detection interval, so this equation is saying that dark noise term is the standard deviation of the mean value in one second (Id) scaled by how long you actually integrate (B).  

My example did not include bandwidth, and so it was not possible to compute a dark noise.  If a bandwidth of 1 MHz was assumed, then 1000 nA mean dark current would yield 0.565 nA standard deviation.  This would pass through the transimpedance amp and become 0.565 uV.  In this case the average dark current is 1770 times larger than the dark noise because each integration period includes 1770^2 = ~3.1 million photoelectrons.