In the Power Functions – Biology!-1 we have seen how unknowingly, we encounter the applications of mathematics in science and in our daily lives. In this article we will the direct implications of the power functions in biology and how it can affect the industrial designing process of the the analytical instruments and the performance.
A student is working in the microbiology lab and wants to separate out the bacterial cells from the suspension. But he is not able to use math and science for choosing the speed/RPM or g value. As he is required to figure out appropriate centrifugal force and separate the bacterial cell mass. He is also unable to decide whether to chose the RPM or g value from settings of the centrifuge. What should one do in such case?
Before we go in details, let us look at some basic physics of the speed of centrifuge. As speed increases, it leads to increase in the mass of the object at speed. Thats why during centrifugation process bacterial cells get heavy and because of the centrifugal force settle down faster at the bottom of the centrifuge tube. After that we remove the liquid part called as supernatant and use the sedimented cellular mass for further experimentation.
Centrifuge rotor
The mass of the single E.coli cell is 1 x 10-12 gm (Reference). under the influence of the earth’s gravitational field denoted as 1g. For faster separation the value of centrifugal force needs to be increased to several thousands depending on the experimental model system (bacteria/animal/plant cells). For pelleting of the bacterial cells the centrifugal force applied is around 10000g for 15-20 mins.
Let’s see how much would be the weight of E.coli cell under this much force. We just have to multiply the weight of E.coli cells with the centrifugal force applied. Which comes to 1X10-8 gm. Under centrifugation the cells get 10000 times heavier compared to their weight under the influence of the earth’s gravitational field and thus separate out of the suspension. If you weigh 65 Kg, imagine that you will be as heavy as 650 tonnes under same centrifugal force used for bacterial separation.
RCF RPM and Radius of the centrifuge – power function
Science math of centrifugation considers two important parameters at the core of its functionality. Revolutions Per Minute (RPM) and Relative Centrifugal Force (RCF) are those parameters. RPM is the speed at which the rotor will rotate and Relative Centrifugal Force is the total centrifugal force times the earth’s gravitational field. The resultant centrifugal force depends upon the RPM and the radius of the rotor, r. Thus to increase the capacity of the centrifuge which parameter between RPM and r can be increased. Let us first see the mathematical relationship between RCF, RPM and r.
RCF= 11.18 X r X (RPM)2 /1000
Let us have a look at RCF values with changing r and RPM values.
| RCF values | R ( at 100 RPM) | RPM (at 10cm radius rotor) |
| 1000 | 8.94 | 95 |
| 2000 | 17.88 | 134 |
| 3000 | 26.83 | 164 |
| 4000 | 35.77 | 189 |
| 5000 | 44.72 | 211 |
Here if we observe carefully we can see that to increase the RCF value from 1000g to 5000g we have to increase the size of the centrifuge almost five times (8.94 > 44.72). This is definitely going to add the significant amount to the cost of the centrifuge. On the other hand we have just doubled the RPM to increase the the Relative Centrifugal Force value five times. Thus it is very much economical to create the centrifuge machines those can rotate faster than making them bigger.
By using simple biomath and calculations companies could save lots of money. But what made the difference. If we look at the relation of Relative Centrifugal Force with r and RPM, we observe that Relative Centrifugal Force is directly proportional to the radius of the rotor but RPM is directly proportional to the square of the RPM value making it a power function. And that’s why small change in the RPM will lead to a significant changes the Relative Centrifugal Force value whereas not much change in the Relative Centrifugal Force will be observed after the increase of radius of the rotor .
This is how practically the power functions from mathematics influence our strategies of scientific methodologies. Analysis with proper methodology is always going to contribute to the scientific and technological enrichment of the society. Thats why learning is important for the betterment. In the next article we will use some biology based examples that requires the use of the mathematical functions. (important resources regarding RPM/RCF conversions)
